*Hi welcome to the first lesson in www.educator.com statistics course.*0000

*Today we are going to talk about descriptive statistics versus inferential statistics.*0005

*Here is the road map for today, first we need to distinguish how statistics is different from other mathematics.*0012

*We will talk about how descriptive and inferential statistics separate.*0018

*Finally we are going to talk about populations versus samples and then we are going to put all of those ideas together *0024

*and look at how population, samples, descriptive, and inferential statistics all fit together.*0030

*First things first, how is statistics different from other specializations in mathematics such as trigonometry, geometry, calculus, linear algebra.*0037

*Statistics is different because it is the science of classifying, organizing, and interpreting or analyzing data.*0048

*You might be thinking to yourself - "Hey science? I thought this was mathematics." Right?*0055

*Its link implies much of science and because of that it is important in mathematics.*0063

*Let me explain that link to you in just one second.*0069

*First I want to step back and think about high school science firmament. *0073

*A lot of high school science is concerned with measurement, we go around measuring things and measuring how fast people run *0077

*and how fast things are dropped and how much things grow and how much things way.*0084

*How big things are and we are gathering a lot of data on measurement.*0089

*Then we find patterns within those measurements and that is basically the fundamentals behind high school science.*0095

*Those patterns can often be described as mathematical formulas.*0104

*I do not know if you have this experience that some of you may have had the experience of trying to derive the gravitational constant.*0110

*To some of you this equation might look familiar, D= ½ gt*^{2}.0117

*(D) stands for distance, (g) stands for the gravitational constant and (t) stands for time.*0126

*Some of you may have had the experience of dropping things off a building and timing them *0138

*and putting in these numbers to try and figure out what (g) is.*0143

*(g) theoretically is supposed to be 9.8 m/sec*^{2}.0149

*But rarely do you calculate exactly 9.8 when you put in distance and time into this equation.*0159

*Often, science students think I'm terrible at science, I’m not getting the right answer *0167

*but it is because all of these measurements are inherently a little bit sloppy.*0173

*Granted that high school students might be sloppier scientists than other scientists but in actuality all science experiments *0178

*have measurement error and there is variance that comes with measurement.*0186

*There is always a little bit of jiggle in that data and often we do not pinpoint the exact right data even when you look at something *0191

*like measuring someone's height, you might have 10 people measure the same person's height and come up with slightly different answers.*0199

*It is not because they are trying to cheat but that person might that a deep breath or slouch a little bit *0207

*or maybe they read the tape measure at their hairline instead at their actual height. *0213

*There are always different reasons for measurement error.*0222

*All science is fought with measurement error.*0225

*While because all experiments, even the good ones at SERV, MIT and Caltech, all experiments will have a little bit sloppiness.*0230

*That is because we are dealing with measuring the physical world.*0242

*It is not bad which we are looking at terrible scientist or just real messy *0250

*it is just that inherently in measuring the world we are going to have a little bit of sloppiness.*0256

*Now because of that sloppiness, even the best experiment will produce a scatter of numbers.*0262

*Even best experiment as well as the worst experiments they will produce a scatter of values or measurements.*0269

*That is where the problem is right?*0289

*You will not get just one number like nice 9.8 gravitational constant, you will instead get this scatter of numbers.*0290

*How do we deal with that scatter and that is where statistics come in.*0299

*Statistics is the math of distributions then you could see how the math part and the science part fit together.*0305

*Statistics is invented because we want to do better in science.*0311

*We even have a special name for the scatter of measurements and that is called a distribution.*0317

*Not only that but we are going to look and see how we can go from frequencies of these values *0330

*in order to get probability distributions of these values.*0337

*Those are also going to be called probability distributions.*0341

*One thing that should come to your mind is that when you have a scatter of values or a whole bunch of different probabilities *0360

*predicting different values then you are not going to have just one number, you are going to have a whole set of numbers.*0366

*Because of that we are going to have to deal with the mathematics a little bit differently.*0373

*We are not just computing one number at a time and looking at one number and adding things to it, subtracting things to it, doing things to it.*0378

*Instead we are looking at entire distributions.*0385

*How do we treat these distributions?*0389

*How do we interpret them?*0390

*That is the question behind statistics.*0392

*You might think working with whole distributions that sounds problematic.*0395

*Sometimes it might seem like it.*0400

*It might seem like these equations are pretty complicated because we have to deal with the whole distribution.*0403

*Also you will get some great stuff out of working with distributions.*0408

*One reason is because distributions are often much more predictable than individual values.*0412

*Distributions are more predictable than individual values.*0419

*Models of distributions or theories of distributions can often predict the mathematical nature of randomness.*0435

*Is it not great?*0444

*They are predicting randomness.*0445

*That is what statistics is a little bit about, it is dealing with that randomness and teaming it.*0448

*How is statistics different from other specializations in mathematics?*0456

*It is born out of the science of classifying, organizing, and interpreting data, distributions of data to be more precise.*0460

*And because of that statistics is the mathematics of distributions.*0469

*Statistics is fundamental in all science in both natural and social sciences.*0474

*I’m a social science professor, a psychology professor by trade but even in the natural sciences all these discoveries that you have heard of *0480

*they only come about through rigorous applications of statistics in physics, biology, economics, psychology,*0490

*you name it statistics have left its math there.*0497

*There are two skills that you need to know when to enter into statistics.*0502

*The first is the skill of data description or what you can think of that as exploration.*0506

*Often you could think of it as just an open-ended examination of the data.*0512

*Let us look and see what is there.*0516

*We are looking for patterns and often it is helpful to make a graph or to look at averages *0518

*and standard deviations that are called summary values when you are looking for patterns.*0524

*These are tools that help us see patterns better.*0535

*The problem with just exploring or describing data is that you are not able to come to any conclusions.*0540

*You have to rain yourself from making conclusions when you are just doing descriptive statistics that is inferential statistics will come in.*0548

*When you make inferences in statistics you are doing a much more strict examination of the data according to set rules.*0557

*Then you will judge whether these patterns that you find through description are likely or not according to theories *0566

*and different models that you may have set up.*0575

*At the end of inferential statistics you should be able to make measured conclusions.*0579

*Often in science we do not say statistics has proven this theory or completely disproven this theory.*0585

*Instead we make much more measured and qualified conclusions.*0593

*Those skills of description and inference applied directly to descriptive statistics and inferential statistics.*0601

*This thing that is different now is you want to think about those skills and how they apply to distributions.*0611

*Here is how descriptive statistics applies to distributions.*0619

*These are the concepts and tools that you need in order to analyze sample distributions.*0624

*Use to describe or explore sample distributions.*0637

*We just have taken the same concepts of what describing data means and we have applied it to sample distributions.*0653

*Distributions that we have plucked out and a set of data that we plucked out.*0660

*In inferential statistics what we need to do is then apply inference to distribution.*0666

*Here it is the concepts and tools to reason from sample distribution.*0674

*To make some inference to reason from a sample distribution to a larger population distribution.*0694

*In inferential statistics what we are doing is using those skills of inference to go from sample distributions *0715

*but not only just to understand the sample but to make some inferences about a greater larger population.*0721

*Just to go beyond our actual data.*0728

*In descriptive statistics we just stay with our sample.*0731

*We do not make any inferences beyond what we have.*0735

*It behooves us to figure out what is the difference between the population and the sample distribution?*0743

*Here it might be helpful to just think of the population a sort of like the truth.*0751

*This is where we are interested in.*0756

*Is it the truth? This is the truth.*0759

*This is the thing that we want to get at.*0765

*If you think about the gravitational constant, this is that magical value that is out there in the world.*0767

*The sample is not the truth, it is like a little bit of that truth.*0775

*When we drop our objects from the top of the building and measure how fast they come down, we are getting samples.*0781

*From those samples we are trying to get at the truth.*0791

*The sample is not the whole truth but the sample does provide a window to the truth.*0794

*It is important to realize that the sample is not the actual truth itself.*0803

*This is not what we want to know about.*0808

*We want to know about the population but we are using the sample in order to know about the population.*0812

*Some pros and cons.*0819

*Some pros of the population is this because it is the truth if you happen to have all the information *0822

*about the real population it will be absolutely 100% accurate.*0828

*However here is the con, it is almost impossible to get.*0836

*It is almost impossible to get the truth, the real population true.*0847

*For instance let us say you just want to know what the real average height of every person in the United States is.*0853

*In order to do that you would have to get measurements from every single person in the United States.*0861

*All of those measurements would have to be 100% accurate.*0868

*Let us say I will give that to you, you will even do that.*0872

*By the time you are finish recording all of those measurements, some people would have died and new people will have been born.*0874

*All of a sudden your measurements would not be accurate anymore.*0881

*It is almost impossible to get the entire population.*0885

*Often in statistics, they will pick a small population like they will say consider all the people who attend your school *0890

*and to shrink down the population that you could think about it without feeling like your mind is being blown.*0897

*In the real world it is basically impossible to get the real truth.*0905

*On the other hand, the sample has the pro of being convenient.*0910

*It is easy to get data from just a sample of the population. *0917

*You do not have to get the whole population, you just have to get a sample of it and it is convenient and easy to get. *0923

*Here is the big con that you need to worry about.*0929

*The con is that the sample might be what is called biased.*0933

*By biased they do not necessarily mean like the sample like racists or prejudiced in some way, *0938

*I just mean that the sample may not be representative of the population.*0944

*The problem with that is when we look at our sample we are going to use our sample to try to get on the truth.*0960

*If our sample is different from the truth then it might lead us astray and that is called being biased.*0965

*When we describe the population in terms of numbers and we get some summary values for the population, *0975

*those descriptive values are going to be called parameters.*0982

*A friend of mine who teaches statistics with a help of the population parameter.*0988

*On the other hand, for samples you would use what is called statistics.*0996

*This word for statistics is the same word as the word for the class.*1006

*But statistics covers all of statistics, descriptive, inferential, population, sample, all that stuff.*1010

*This is the sort of smaller use of that word.*1018

*Population and parameter, specific sample for statistics.*1024

*Now let us put all those ideas together.*1033

*How do we put together descriptive and inferential statistics with populations and samples?*1036

*It helps us to ground ourselves by starting off with the idea that what we are interested in, in knowing about is the entire population.*1042

*We want to know about the real population.*1052

*Let us deal with one population at a time for now.*1056

*Often we do not have the population's entire data in front of us, we only have a sample of that data. *1060

*This is our wish to go from sample to the population but remember the sample can be biased, that is problematic.*1069

*Here is where statistics comes in.*1080

*From samples we compute statistics and from populations we could know the parameters.*1083

*But we often do not have this link either because we do not know anything about the actual population.*1097

*Here is where we are, what inferential statistics will help us do is make this link.*1106

*How do we go from statistics of the sample to population parameters?*1114

*This jump, this inferential jump is going to be made through inferential statistics.*1119

*However in order to go from the sample to statistics we will use descriptive statistics.*1134

*This is how it all fits together.*1147

*Let us try some examples. *1150

*Here is example 1, a pollster asks a group of voters how they intend to vote in the upcoming election for governor.*1153

*In this example is the individual pollster primarily using descriptive statistics or inferential statistics.*1161

*What he or she computes parameters or samples.*1171

*Here the pollster is just asking a group of voters how they intend to vote.*1175

*A poll is often just a sample of the entire set of voters so I would say the pollster is probably going to compute some sample statistics.*1180

*We should say statistics not samples.*1194

*I would say the pollster is probably calculating statistics.*1202

*If the pollster just got an answer such as this sample of voters is going to vote for the governor 75% of them are going to vote for the governor *1208

*and only 25% are not that would be counted as descriptive statistics.*1219

*Once this pollster actually uses that information to then make some inferences and predicts and then I predict the governor will win, *1225

*that would be inferential statistics.*1236

*But so far, it does not say that.*1238

*It seems that only descriptive statistics is being used here.*1242

*Example 2, a teacher organizes his classes test grades into distribution from best to worst and compares it to the test grades of the entire school.*1248

*In this example is the individual primarily using descriptive statistics or inferential statistics.*1259

*First he is definitely using descriptive statistics in order to organize his classes data.*1265

*He is using this but then he is comparing it to the test grades that the entire school.*1273

*He is getting his sample, his class and looking at how they are relative to the entire school.*1279

*That leap is going to be inferential statistics.*1290

*I would say he is using both descriptive and inferential.*1294

*A statistician is interested in the choices of majors of this year’s entering freshmen at a university 10% of randomly sampled.*1302

*What is the population? what is the sample? What is the parameter? What is the statistic?*1311

*The population seems to be all freshmen at the University, right? but the sample is this 10%.*1317

*That is the population and the sample so what is the parameter?*1337

*The parameter is what are the real major choices of all the students.*1342

*Maybe he will look at it as you know maybe 50% are engineering and 20% are science and 30% are humanities.*1355

*Majors picked by freshmen.*1374

*What is the actual statistic?*1383

*The statistic that is going to be made up of the majors picked by the sample.*1386

*In order to go from this to this, you will need to use inferential statistics.*1401

*Example 4, a group of pediatricians are trying to estimate the rate of increase in obesity in young children in their city.*1410

*They begin a research project for every four years a group of 8 year-old children are randomly sampled from the city and weighed.*1418

*What is the population? What is the sample? what is the parameter? what is the statistic?*1425

*The population looks like young children in the city, whichever city this happens to be.*1431

*The sample is the group of 8 year-old children, group of selected to be in this study.*1446

*What is the parameter? *1469

*The parameter would really be the actual rate of increasing obesity and they do not know what that is, they can not get that data.*1474

*By looking at the different groups of 8 year-old children every four years they could look at the rate between the samples.*1490

*The statistic would be the rate among the sample, the samples every four years.*1503

*In that way they will try to use this rate in order to estimate this rate.*1521

*That is the end of lesson one for www.educator.com.*1527

*Thanks so much for watching.*1530

*Hi and welcome to www.educator.com.*0000

*Today we are going to be introduced to competence intervals.*0002

*Here is the roadmap for today, first we are going to do a brief overview of inferential statistics.*0005

*We have been trying to do some inferential statistics but there have been a couple of problems we keep running into.*0013

*So far I have fudged it.*0022

*We will address some of those problems head on and come up with 2 solutions.*0024

*One of those solutions is the competence interval and we are going to talk about competence intervals *0031

*when the sigma, population, standard deviation is known and when sigma is unknown.*0039

*Those are the two situations we are going to be focused on.*0046

*Let us go over inferential statistics.*0049

*We know the big picture idea there is some population represented by X and we wish we could know the population but we do not.*0055

*But instead what we can know is little samples.*0065

*We could know that but the problem is samples are biased. *0071

*Whenever we have samples and we summarize them using these mathematical summaries we call them statistics.*0074

*Just to give you an example of some statistics there things like x bar or s, those are all statistics.*0084

*What we would like to do is use these samples to understand something about the population.*0093

*Statistics, the field is about using these statistics to estimate parameters and *0100

*to give you ideas about parameters there are things like mu or sigma.*0108

*That is our whole goal.*0112

*Here we realize in order to jump from things like x bar and s to mu and sigma we are going to need more than just wishful thinking.*0114

*And that is where the sampling distributions come in.*0132

*Here we talk about sampling distribution often we are talking about some sort of statistic.*0135

*When we talk about sampling distribution of the mean we are talking about a whole bunch of x bars.*0142

*Here we have a whole bunch of x.*0148

*Here we have a whole bunch of x bar and that is the distribution.*0150

*When we summarize these statistics in the sampling distribution we call them expected values. *0155

*So it is not just mu, it is mu sub x bar. *0164

*It is not just sigma it is sigma sub x bar. *0168

*What we want to do is go from this to understand this but what we have learned *0172

*so far is how to see the relationship between parameters and expected values.*0177

*We know that these things have a relationship to each other.*0186

*And from doing that we could then make this jump.*0189

*It is like we use this to say something like this.*0195

*There are two problems with this picture although it seems rosy and there is still to nagging questions.*0199

*We would look at them a little bit before but we need to solve this more rigorously than we had before. *0210

*One question is this, what happens when we do not know what the population looks like?*0217

*Of course we could use the central limit theorem when we know mu and sigma from the population.*0222

*What if we do not know mu?*0229

*What if we do not know sigma?*0231

*Then what happens?*0233

*Also, how do we know whether a sample is sufficiently unlikely because remember the whole point *0234

*of the sampling distribution is for us to take sampling distributions from a known population and compare it to an unknown population. *0240

*If this sample does not match the sampling distribution enough that it is very unlikely to come from the sampling distribution.*0254

*We could say this is probably not the population that the sample came from.*0261

*How do we know when it sufficiently weird?*0266

*To answer these two questions there is going to be to solutions.*0269

*You can think of it as this one.*0275

*This first question roughly, they are both actually are answered in each of these but this one goes along better with that one.*0281

*This one goes along better with that one.*0287

*The two solutions are these, one is competence interval.*0291

*When we talk about competence interval here is what we are doing, we are going to figure out where mu might be from the sample.*0302

*We are going to try to figure out the population mu from the sample and *0306

*that is what we do when we do not know what the population looks like.*0336

*We try to figure it out from the sample.*0342

*Hypothesis testing actually takes another view.*0344

*The hypothesis testing, we come up with a hypothesis for what the population is like.*0349

*Hypothesize a population mu first.*0355

*In this case we are saying we are going to pull from something and figure out and pick a potential population mu.*0363

*And then we are going to test how weird the sample is.*0376

*We are going to come up with a number to tell us this is how weird the sample is.*0387

*We are going to decide is that weirdness weird enough?*0393

*That is going to be hypothesis testing.*0398

*But we are going to focus here on competence intervals. *0401

*Okay, when we talk about competence intervals we need to get an inventory of what we know so far.*0404

*Basically that is asking the question, which parameters are known or given to us?*0413

*What happens when we do not know what the population looks like?*0418

*Well we may not know what *0422

*The population looks like because we do not know anything about the population, or we know *0424

*Only a little bit about the population.*0428

*This is the case where we know a little.*0431

*Here we do not know mu but we do know sigma.*0434

*For some reason we have some partial information and that helps us out.*0444

*Here we know nothing.*0450

*Here nothing is helping us that we do not know mu and we are trying to figure it out but we do not know sigma either.*0454

*It is like nothing is helping us out here.*0464

*We just have to pull ourselves up from our own bootstraps.*0466

*These are the two situations that we are going to talk about it.*0471

*Here is the goal of competence interval. *0475

*The basic idea of the competence interval is going to be this.*0480

*We are going to try to figure out where mu might be but we do know x bar.*0484

*We know everything about the sample but we do not know anything about the population.*0497

*But in this case I am going to show you what happens when we already know sigma.*0503

*So we have a leg up. *0508

*We know sigma life is little that easier for us today.*0509

*Here is the thing, we do not know what the population looks like so cannot draw a normal or skewed or anything. *0513

*We have no idea what the population looks like and we have no idea what the population mu is.*0524

*But we for some reason know sigma is.*0531

*Sigma is given to us.*0533

*From there can we construct an SDOM?*0534

*Given that n is sufficiently large we can assume that it is normal. *0540

*We have no idea what mu is and so we do not know what mu sub x bar is. *0548

*We do not know it at all but we can figure out sigma sub x bar.*0553

*We could figure out the standard error because we have sigma and we could divide that by √n.*0559

*We have a little bit of information about the SDOM.*0566

*Here is what we do in competence intervals.*0570

*First assume that the x bar is the mu sub x bar.*0574

*Whatever your sample x bar is we are going to put back here.*0586

*We are going to assume it.*0591

*Here is why, because we always assume one thing to figure out the other, *0595

*here we are going to assume things about the x bar to figure out mu.*0601

*And hypothesis testing, we assume something about the population to figure out how *0605

*Weird x bar is.*0609

*Here because we know that the SDOM tends to be normal given a sufficiently *0612

*Large n what we know is that we can find out with reasonable competence what some *0621

*Significant borders are.*0632

*For instance, let us say we are one standard deviation away.*0634

*This is raw score and this is z score so we know at one standard deviation away *0642

*this base right here we know that that is 68% of SDOM.*0650

*Let us think about what this might mean.*0660

*When we get these borders what we might end up saying is that these are the borders in which 68% of our values will fall in the SDOM.*0663

*And here is what we could say we could also say that there is 68% chance that our *0679

*Population mu will fall in that zone.*0686

*That is a 68% competence interval.*0691

*For 68% is higher than half, but it is not that high.*0697

*But here is the thing we can have a high competence interval.*0702

*We can have a 95% competence interval or we can have a 99% competence interval.*0707

*That is what we can do. *0713

*We can have here is my x bar, here is 0 but what we can do is figure out *0716

*These borders such that we are now sure that 95% chance of having our *0730

*Population mean fall in this interval.*0744

*We can know that.*0748

*That is called the competence interval.*0750

*That is pretty hypothetically and you can even go to 99%.*0753

*And we could easily figure out these borders. *0756

*Here is how.*0759

*Because we easily figure out the border we could figure out what the z scores are.*0761

*This is what we call a two-tailed competence interval because even though the middle part is 95% that does not mean that part of 5%.*0772

*You will have 105% so that means that part is .025 so 2.5% and this part is .025.*0785

*And those the only parts that we are not sure.*0796

*There is a small chance that the population mean will fall somewhere out here but it is a very small chance. *0798

*We are trying to reduce it as much is possible.*0809

*Let us think about how we could find the z score out here.*0812

*We could use our tables in the back of the book, our z tables and we can look up and usually z tables will give you like one side.*0817

*We can look up .025 and look at the z score or we could do it on our Excel.*0830

*Instead of using normsdist, normsdist will give you the proportion of the distribution.*0837

*We are going to put in normsin as the inverse and here we want to put in the probability.*0848

*Now this is going to be my probability.*0855

*I am going to put in this probability .025 and we get 1.967.*0870

*This value here is -1.96 and because the normal distribution is symmetric we know that this part is also 1.96 *0884

*but now a positive instead of negative.*0892

*We know our z values on the end and if we know the z values what is our raw score here?*0896

*Tell me what this value is and also tell me what that value is.*0908

*Well the z score tells you how many standard errors away you are.*0915

*How many jumps away and each jump is worth that much.*0921

*We are away 1.96 of these jumps.*0926

*We are going to multiply this by this and then *0931

*Either subtract it from x or add it to x.*0934

*Step two in finding competence interval is let us say you want to find a 95% competence interval finds the z scores.*0938

*It is all in the case where you know sigma.*0953

*Step 3 is this, now you want to actually find the actual scores and that is going to be x bar + or -the z score × standard error.*0957

*That is what you are going to do. *0984

*And we know what the standard error is.*0986

*I am going to rewrite this to be x bar + or - z score × sigma / √n.*0989

*When we do that we could find these competence intervals.*1003

*Once you have these competence intervals then you that with 95% competence that *1009

*your population mean will fall in this interval between these two numbers.*1019

*Now the 95% is actually called the capture rate that is like 95% and 99%, whatever. *1028

*What would the competence interval be for 100%?*1042

*It would go from –infinity to infinity because that is how far the normal distribution goes.*1047

*But the capture rate is this the proportion of random sample for which this interval captures mu.*1053

*Let us imagine taking a whole bunch of random sample, it is going to be that 95% of the *1080

*Time those random samples in tail mu.*1091

*They somehow overlap with mu.*1097

*That is what we mean by 95% capture rate.*1099

*That is when you know sigma but now we do not know sigma.*1103

*We are in trouble but we do not know mu.*1113

*We do not sigma either.*1115

*Still our goal remains the same, we try to figure out mu from x bar.*1116

*But now we are a little hobbled.*1128

*I do not have a tool that I use to have.*1132

*The beginning part of the story stays the same. *1135

*The population we have no idea and from there we want to find the SDOM because *1139

*we are going to figure out how good our sample is.*1146

*We know the shape of our SDOM as long as our s is sufficiently big.*1151

*Can we figure out sigma sub x bar anymore?*1157

*No we cannot because we do not have sigma so how can figure out sigma sub x bar.*1161

*We cannot figure out that standard error.*1170

*Here is where another idea comes in.*1171

*There is another way we can estimate the standard error of the sampling distribution that is going to be s sub x bar.*1175

*Because we are going to use the sample standard deviation s instead of sigma.*1186

*Remember s is more variable, not quite right and because of that we corrected already a little bit by using n -1 instead of n.*1200

*Here we are going to divide that by √n.*1214

*If you double click on this you would see the square root of the sum of squares ÷ √ n -1.*1218

*You would see this inside of that.*1231

*We already tried to correct it a little bit, but s is still variable.*1234

*It is not quite as good as having sigma.*1242

*And there can be other problems that we run into.*1245

*This is pretty good though and it is a pretty good estimate but you always have *1249

*to keep in mind we have not as good of a standard error as we used to.*1254

*We have to account for that.*1262

*But the steps remain the same. *1265

*First assume x bar for mu sub x bar. *1267

*Two, find z for your capture rate.*1275

*If your capture rate for example 95% then you would find the z scores.*1287

*It is helpful to memorize that for this capture rate the z scores are going to be + or -1.96. *1297

*It is going to come up a lot.*1305

*Find the z scores for your capture rate.*1306

*Here we run into a problem. *1310

*I wish we could use z scores but here is an issue, we actually cannot because s is to variable for us to assume perfect normality.*1314

*And because of that we cannot use the z and instead we have to use the t which is very similar to z.*1330

*Find the t score for your capture rate.*1348

*Instead of having raw score and z score we are going to find t score.*1352

*For now you just need to know that you can find your t score in the back of the book but in *1366

*The next lesson we are going to go over why you use t and why you cannot use z.*1372

*That is a big story.*1377

*You are going to find t.*1380

*Once you find the t for your capture rate and that will also be + or -, t is going to be very similar to z score.*1383

*We are going to use this formula.*1390

*You are going to use a very similar idea to the z score competence interval where you want to know x bar + or -.*1396

*How a t score is also going to tell you how many standard errors away.*1407

*T × standard error. *1411

*But remember, you use t when you estimate this from sample.*1417

*If we unpack this, this is what it can look like x bar + or - t × this is that estimated standard error s/√n.*1426

*It is still the same idea.*1443

*It is how many jumps away, figuring that out and then multiplying that to the length of the jump *1446

*and adding that to x bar for the high-value and then subtracting that from the x bar for the low value.*1451

*In order to find t here is what you need to know for now.*1458

*You need to know whether it is a 1 or 2 tailed distribution.*1465

*If your competence interval is two-tailed then remember these are .025 *1470

*because you would split the remaining 5% on both side.*1478

*But sometimes where t values though only give you one side.*1482

*They might give you a one sided 5% or one sided .25%. *1487

*You have to just keep in mind whether it is one tailed or two tailed and also the t distributions are a whole bunch of different distributions.*1493

*They are a whole bunch of different tables basically.*1502

*You have to also know what degrees of freedom.*1508

*For now you could remember degrees of freedom as n -1.*1514

*There are reasons for all of these things why we use t, why we use degrees of freedom all that stuff.*1521

*That will be covered in the next lesson. *1528

*For now, here is what you need to know.*1529

*You need to know whether it is one tailed or two tailed.*1532

*You also need to know degrees of freedom. *1534

*Once you have that you could actually look it up in t table usually found in the back of your book. *1536

*It might also be called the students t distribution because - invented it but he was actually contracted to work for Guinness.*1542

*That is why I cannot publish it under his actual name.*1553

*We published it under the pseudonym student because that is called the students t.*1556

*You can look up your degrees of freedom and then look for the area that you need and then go down and find the t score.*1560

*Very similar to z score.*1573

*Let us go on to some examples.*1574

*Example 1, consider two extreme situations n=10 and n=1,000.*1582

*If you use s in the formula for CI given sigma, here is the actual formula for when you have sigma.*1591

*We use 1.96 because we use the z score.*1609

*Which of these situations would you expect to give a capture rate closer to 95%?*1614

*Here is what this question is really asking.*1621

*When you know sigma for competence interval for 95% competence interval 1.96 that is my z × sigma / √n.*1624

*What it is asking you is what if you substituted in s?*1649

*Here we do not know sigma but we are going to just take this formula and use the z value s/√n.*1656

*In order to answer this question you really only need to keep in mind one thing, when is s more like sigma.*1676

*S is more like sigma when n is very large.*1687

*This situation would give you a very close capture rate of 95%.*1708

*This would be very, very similar. *1721

*However, when n is 10 you have more uncertainty and because of that the t distribution it is not as tight.*1724

*It is actually more like spread out and because of that, when n=10 you do not capture 95% just by being about 2 standard deviations out this way. *1733

*That would not capture 95% of those samples.*1748

*In fact you have to go out further to capture 95%.*1753

*This is going to be much closer to 95% capture rate.*1758

*This is going to give you a smaller capture rate.*1763

*That is because your s is going to be more variable and because of that your t distribution *1766

*is going to be more disperse because more variable means sort of wider.*1778

*95% CI for a population mean is calculated for random sample of weights and the resulting CI is from 42 to 48 pounds.*1785

*For each statement indicate whether it is a true or false interpretation of the CI.*1798

*This question is asking you do you understand what the competence interval means?*1807

*Do you understand what it is for?*1811

*Let us see, 95% of the weights in the population are between 42 and 48.*1813

*Does competence interval tell us about the actual population numbers?*1821

*No, it only tells us about the population mean.*1830

*This is actually not true.*1833

*We do not know anything about the actual numbers of the population. *1836

*We do not know whether it is skewed, whether it is uniform distribution.*1840

*We do not know any of those things. *1847

*The 95% thing would only be reasonable if the population was normal and its mu was exactly equal to x bar.*1848

*That would be the case.*1862

*That is not true.*1864

*What about number 2?*1866

*95% of weights in the sample are between 42 and 48, does the CI tell us anything about this sample?*1868

*No, using the sample to estimate population mean.*1878

*We are using the SDOM.*1882

*We do not know anything about the sample itself.*1884

*That is also not true. *1888

*What about number 3?*1890

*The probability that the interval includes the population mean is 95%. *1893

*This is actually true. *1899

*There is only a 5% chance that this interval does not contain the population mean.*1902

*What about number 4?*1916

*The sample mean might not be in the competence interval.*1919

*That does not make sense if you look at the picture because we use the sample mean in order to construct the competence interval.*1924

*Of course this is in the competence intervals and this is just ridiculous. *1932

*Example 3, a random sample of 22 men had a mean body temperature of 98.1°, standard deviation of .73.*1936

*Construct a 95% competence interval for the mean of the population that the sample was drawn from.*1950

*Interpret the CI and 98.6° included in this.*1956

*This the average human body temperature.*1963

*We have body temperatures in the world and we do not know what that population looks like.*1965

*We are asking can we construct 95% competence interval such that whatever *1975

*the population mean is there is a 95% chance that we have covered it.*1989

*We start by assuming that the mean of the sample x bar is the mean of the sampling distribution of the mean.*1994

*We have done step one.*2004

*Step two is we have to construct CI and so here they give us x, but do we have sigma?*2008

*No.*2023

*We know that we cannot use the z score.*2025

*We have to use the t score. *2029

*Let us find the t for this.*2031

*This is .025 chance that we would not find it on the site and here is .025 chance that we can find it on the site. *2033

*What is the t scores?*2043

*This is the raw score or the temperature. *2046

*What is the t score for .025 when the degrees of freedom and that is n -1 there is 22 man so 22-1= 21 degrees of freedom. S*2049

*If you look in your book, at your students t distributions I am going to go down to where the df=21.*2065

*I am going to go across to where it says you know .025.*2074

*My table actually gives me this area so I am going to look at .025 on the side.*2080

*You and it says 2.08 is my t score.*2086

*That makes sense.*2093

*That is around 1.96.*2095

*You will see that as degrees of freedom get greater and greater this value becomes more and more close to 1.96.*2098

*On this side we know that it is symmetrical so I know it is -2.08.*2108

*From here I can construct my CI.*2114

*The CI is going to be the x bar + or – the t value × my standard error.*2118

*My estimated standard error here is s sub x bar because we do not have sigma.*2129

*That is going to be s ÷ √n.*2137

*Let us put in numbers here, so that is 98.1 that is our sample mean ± t value 2.08 × s .73 ÷ √22.*2141

*I am just going to calculate this on a calculator so that is going to be 98.1 and I will do the + side first. +2.08.*2167

*Excel does order of operation.*2182

*It needs to do the multiplication before the addition and its .3 ÷ √22.*2185

*That is the high-end of my competence interval is 98.4 and the low end is going to be 97.8.*2195

*98.4 and 97.8 are my CI.*2217

*When we interpret the competence interval we want to say something like *2229

*there is a 95% chance that the mean of the population lies between these two values.*2239

*Or another way we could say it is that if we draw samples at random, 95% of those samples will include the population mean.*2250

*95% of the samples in between this interval will include the population mean.*2264

*Let us think about this competence interval, is it reasonable?*2271

*Is 98.6° included that is supposed to be the mean for everybody.*2280

*We see that it is not actually.*2286

*Maybe this sample is odd because our competence interval does not actually include the mean *2288

*that we secretly know for providing temperature of people.*2297

*That is when competence intervals are helpful. *2307

*Here is example 4, in a random sample of 1000 community college students, their mean score on a quantitative literacy test was 310.*2310

*The standard deviation on this test of all the community college students have taken is 360.*2324

*Construct a 95% competence interval for the mean of all community college students have ever taken this test.*2331

*Here is our random sample and their mean or x bar is 310 but the standard deviation *2338

*of all the students who have taken this test that is the sigma is 360.*2351

*Construct a 95% competence interval. *2358

*Well, the first part that we know population we do not know but we are given the population standard deviation.*2361

*And from that, let us construct the SDOM.*2374

*Well given that this n is quite large let us assume normality.*2377

*Here we could find out the standard error by putting 360 ÷ √ 1000.*2382

*Now going to our steps of our competence interval first we assume that x bar is the mean of our sampling distribution of the mean.*2395

*Here we could use the z instead of t because we have sigma and because of that we know that this is normal. *2412

*That is going to be +1.96 and -1.96 in order to construct a 95% competence interval.*2425

*Our CI is going to look something like this x bar + or – z × standard error.*2436

*If you sort of double click on standard error what you will find is sigma / √n.*2446

*Let us put in numbers here.*2464

*310 is our x bar.*2467

*Our z score is 1.96.*2471

*Our sigma is 360.*2475

*Our n is 1,000.*2479

*Let us put these in our calculators.*2483

*I will do the high end first 310 + 1.96 × 360 ÷√1,000.*2487

*Order of operations says it does not matter anything you multiply or divide it in.*2508

*That is my high end 332 as the high scoring end.*2516

*The low scoring end, the lower bound of my 95% CI is 287.7.*2524

*That is going to be 287.7 as well as 332.3.*2537

*The mean of the population 95% should fall between this interval.*2547

*That is the end for our competence intervals.*2558

*That is part one of competence intervals.*2561

*Hope you join me for t distributions to find out why we use t instead of z sometimes.*2566

*Thank you for using www.educator.com.*2571

*Hi and welcome to www.educator.com.*0000

*Today we are going to talk about t-distribution.*0001

*Previously, we learn that there are different situations where we use z and when you use t.*0004

*Today we are going to talk about when to use z versus t.*0011

*We are going to break down and sort of reflect and recognize what is z and t?*0015

*What do they have in common and with is different about them?*0022

*For certain cases we are going to ask question, why not z why t instead?*0024

*What does not z have?*0031

*What is deficient about z? *0033

*We will talk about rules of t distribution, they follow certain patterns and t distributions *0035

*are a family of distributions separated by degrees of freedom. *0044

*Different t distributions have different degrees of freedom.*0049

*We are going to talk about what are degrees of freedom?*0053

*We are going to talk about how degrees of freedom relates to that family of t distribution, and then finally summarize how to find t.*0056

*First off, when do we use z versus t?*0065

*We covered in the previous sections where we look at whether we knew the population parameters or not.*0072

*In hypothesis testing, we frequently do not know the mu of the population, but sometimes we are given sigma for some reason or another. *0080

*In this case we use z in order to figure out how many standard errors away from the mean we are in our SDOM.*0091

*But in other situations, we do not know what sigma is.*0102

*In that case we use t in order to figure out how many standard errors away our x bar is from our mu.*0107

*Just to draw that picture for you remember we are interested in the SDOM because the SDOM tends to be normal given certain conditions.*0118

*Although mu sub x bar = mu given the CLT what we often want to know is if we have x bar that fall here or x bar that falls here.*0126

*We want to know how far away it is from the mu sub x bar.*0147

*In order to find that we would not just use the raw score and get the raw distance but we would want that distance in terms of standard deviation.*0153

*But because this is the SDOM, we call it the standard error.*0165

*We would either want a z or t and these numbers tell us how many standard errors away we are from this point right in the mu.*0168

*What is the z and t?*0181

*The commonality as we saw before is it tells us number of standard error away from mu sub x bar and that is common to both.*0186

*That is what the z score and t score both have in common.*0208

*Because of that their formula looked very much the same. *0213

*For instance, one way we can write the z formula is like this. *0217

*We have x bar - mu or mu sub x bar they are the same and this gives us the distance in terms of just the raw values.*0231

*Just how many whatever inches away, points away, whatever it is.*0251

*Whatever your raw score means, degrees away divided by standard error.*0258

*If we double-click on that standard error and look at what is inside than the standard error also written as sigma sub x bar *0264

*because it is the standard deviation of a whole bunch of mean = sigma ÷ √n.*0275

*If we look at the t score formula then we have almost the same formula.*0284

*We have that distance ÷ how big your little steps are, how big your standard deviations are.*0294

*But when we double-click on the standard error like something on the desktop, you double-click it and open it up what is inside?*0302

*Well, you could also write this one as s sub x bar and that would be s ÷ √n.*0311

*Here in lies this difference right there.*0323

*That is our difference.*0325

*Here the difference is that standard error found using the sigma, the true population standard deviation.*0327

*Obviously if you use the real deal that is better or more accurate than the standard error found using estimate population standard deviation.*0345

*That is s.*0371

*S is estimated from the sample and if we double clicked on s it would look like this.*0375

*It is that basic idea of all the squared deviations away from x bar, away from the mean of the sample.*0383

*X sub i - x bar*^{2}.0395

*We have all the squared deviations and we add them up ÷ n -1 because this is our estimate of the population standard deviations *0402

*and all of that under the square root sign in order to just leave us a standard deviation rather than variance.*0414

*This is an estimate of population standard deviation. *0421

*It is not the real deal, so it is not as accurate. *0426

*One thing you should know is that the z score is less variable and the t score is going to be more variable.*0430

*That is going to come in to bear on why we use which one. *0438

*Okay, so why not z?*0443

*When we have situations where we do not have the population standard deviation, why not z?*0448

*Why cannot just be like you are using s, why cannot we do that?*0458

*Why do we use t? *0466

*It is because we use s this is actually something a little bit weird.*0468

*The weirdness comes from the fact that this s is much more variable than sigma.*0474

*Sometimes when we get our estimate, our estimate is scat on.*0481

*Sometimes when we get our estimate it is off.*0485

*That is what we mean when it is more variable.*0489

*It is not going to hit the nail and head everything single time.*0491

*It is going to vary in its accuracy. *0495

*Now z scores are normally distributed when SDOM is normal.*0497

*Here is what this means.*0502

*The way you can think about it is like this, when the SDOM is normal and we pick a bunch of points out *0503

*and find the standard error from those points and plot those, we will get another normal distribution.*0516

*But that is not necessarily the case for s.*0523

*Here we need to know that z scores are nice because z scores is going to perfectly cut off that normal distribution accurately for you. *0530

*Remember, the normal distribution it always has that probability underneath the pro and it has these little marks.*0547

*These can be set in terms of z scores.*0557

*What is nice about the SDOM when it is normal is that when we have the z score it will perfectly match to the proportion of the curve that it covers.*0563

*This will always match.*0579

*The problem is t scores do not match up in this way.*0581

*We can just say why do we just call a t score a z score and still use the same areas underneath the curve?*0587

*We cannot do that because that is just the superficial change.*0600

*Here is what we mean by the z scores are normally distributed. *0603

*When you get z scores and when we talk about normal distribution, I'm not just talking about that bell shaped curve.*0611

*Yes overall it should have that bell shaped general shape but it is a little more specific than that.*0619

*You can have the bell shaped and have the perfect normal distribution.*0628

*For instance, 1 standard deviation away this is area will give you 34% of the area underneath the curve.*0635

*This area is about 14% and this area is about 2%.*0645

*That is a true normal distribution. *0653

*This on the other hand, it looks on the surface as if it is normally distributed. *0656

*It looks like that bell shaped curve, but it is not. *0662

*Here is why.*0665

*This area, I should have actually drawn it a little bit differently, but I want to show you that do not go by appearances.*0666

*Appearances can be deceiving. *0677

*This might actually be a little bit less than 34%.*0678

*It might be something like 25%.*0685

*If that was the case, you would see this area and that area is not 34%.*0688

*It is 25%.*0700

*Not only that but this area is now a little bit more than 13 ½, it is around 14%.*0701

*Now this area is not 2% but 11%.*0710

*Although it looks like a bell shaped curve, it is not quite a normal distribution because *0715

*it does not follow that empirical rule that we have talked about before.*0722

*What is nice about z scores is that z scores will always fall in this pattern. *0726

*These z scores will always correspond to these numbers.*0731

*That is why you could always use that z table in the back and rely on it.*0735

*The t scores are not going to do that for you.*0739

*T scores may not give you that perfect 34, 13 ½ and 2% sort of distribution. *0746

*Even though the SDOM might be normal, the t scores are not necessarily normal.*0753

*We had this normal thing and we have t scores and how do we go from t score's defining this area underneath the curve.*0762

*That is the problem we have here.*0772

*It turns out that if n is big then this does not matter as much. *0774

*It n is really large, if your sample size is large then the t distribution approximates normal. *0782

*It goes towards normal but when n is small, then you have to worry.*0788

*Also when n is in the middle or when n is big, it is just large.*0795

*There are all these situations where you have to worry about the t as well as the area underneath the curve.*0801

*If the t scores are not normally distributed then we cannot calculate the area underneath the curve.*0810

*If we have our lovely SDOM and we know that the SDOM is nice and normal and we have our mu sub x bar here then everything is fine and dandy.*0816

*We have x bar here and we want to find that distance, and we find the t score.*0832

*The problem is we cannot translate from this directly into this area.*0838

*That is the problem we ran into.*0844

*Here what we see is this sort of more like a t distribution than a z distribution.*0847

*I'm just going to call the z distributions to call them basically, the normal distribution.*0865

*The t distribution is often a little bit smooched.*0871

*Think of having that perfect normal bell shape.*0876

*It is squishing the top of it down.*0880

*It makes that shape ball out a little bit.*0882

*It is not as sharply peaked but a little bit more variable.*0888

*We had said the s is more variable than the sigma.*0895

*It makes sense that the t comes from s is more variable than the z that comes from sigma.*0902

*You might be thinking what are we stuck?*0911

*We are not stuck and here is why.*0921

*He actually worked out all the t distributions as well. *0924

*He manually calculated a lot of the t distributions and made tables of the t distributions that we still use today. *0928

*He published those tables and under the pseudonym the student.*0944

*At the time he was working for Guinness brewery and he could not publish because they were sort of like we do not want you to know who we are.*0949

*Our secrets are very dark beer.*0957

*He published under the pseudonym and because of that some of your t distributions *0960

*in the back of your book maybe labeled the students t to talk about Bosset’s t.*0967

*Here is what Bosset found, he found that t distribution can be reliable to.*0973

*You can know about them it is just that you need more information when you need for the z distribution.*0980

*For z distribution you do not need to know anything.*0988

*You just need to know z and it will give you the probability.*0990

*Life is simple.*0993

*T distributions are not that simple, but not that complicated either.*0995

*They had a few more conditions to satisfy and the biggest condition that you will have to know is about degrees of freedom.*1002

*Because for each degree of freedom there is a slightly different t distribution that goes along with it.*1012

*Let us talk about some of the rules that govern t distributions.*1024

*The first one you already know as t distribution gets more normal as n gets bigger.*1031

*This makes sense if we step back and think about it for a second.*1039

*Imagine if n=n then what would your s be?*1042

*If your sample is like a entire population then s should be much closer to the actual *1054

*population standard deviation much better than when n is small.*1071

*It is still a little off because of the n-1 thing but it is very close and that is the closest you can get.*1077

*T distributions are more normalized and gets bigger because s is a better estimate of sigma as n gets bigger.*1085

*That makes sense.*1111

*The problem all stems from s.*1113

*It is variability that as s gets better, less variable and more accurate to the population then t gets better.*1116

*T is based on s.*1128

*That is like t distributions are more normalized as n gets bigger.*1130

*T distributions are a family of distribution.*1135

*It is not just one distribution.*1138

*It is a whole bunch of them that are alike in some way and it depends on n.*1140

*It depends technically on degrees of freedom, but you can say it depend on n sometimes because degrees of freedom is often n -1.*1145

*There are other kinds of degrees of freedom this is the one you need to know for now. *1154

*But later on we will distinguish between different kinds of degrees of freedom.*1159

*Degrees of freedom is actually important as a general idea it is just the number of data points -1.*1163

*We have a family of distributions. *1174

*They all look sort of a like.*1178

*They are all symmetrical and they are unimodal and they have that bell like shape, but they're not quite normal. *1179

*Not all of them.*1190

*As n gets bigger, or as degrees of freedom gets bigger the distribution becomes more and more normal.*1191

*Let us step back and talk a little bit about degrees of freedom first.*1201

*Let us assume there are three subjects in one samples so n=3.*1207

*We know that just by the blind formula n -1 degrees of freedom is 2 but what does this mean?*1213

*Here is the thing.*1224

*Let us assume there are three subjects in one sample and let us say it is some score on a statistics test.*1228

*They can score from 0 to 100 and if I say pick any 3 scores you want and that could be those subject scores.*1235

*Your degrees of freedom would be 3.*1244

*You are free to choose any 3 scores.*1246

*You are not limited.*1249

*You are not restricted in any way.*1250

*If you figure out any sample statistic, let us say the mean or variance.*1253

*If you figure out any sample statistic then if you randomly picked 2 of those scores you can no longer just pick the 3rd score freely.*1261

*You have to pick a particular score because you already used up some of your populations for the mean.*1274

*The mean will constrain which two scores you could pick.*1283

*This logic will become more important later. *1288

*Let us put some numbers in here.*1292

*Let us talk about the case when n= 3 and degrees of freedom = 3.*1294

*It would be like there are three subjects and they could score from 0 to 100.*1299

*I am totally free.*1310

*I can pick 87, 52, my last score I can pick anything I want.*1314

*I can pick 52 again, 100, or 0.*1321

*It does not matter.*1325

*I can just pick any score I want.*1326

*If I erase these other scores I will just put in a different score.*1328

*It does not matter. *1333

*I'm very free to vary.*1335

*But let us talk about the most situations that we have in statistics where we figure out summary statistics. *1337

*Here we have n=3 and degrees of freedom =2.*1345

*Here is why.*1350

*The score is the same, it can go from 0 to 100.*1351

*We also found the x bar =50.*1358

*If we found that the x bar = 50, then we cannot just take any score all 3 times.*1363

*Can we pick any score for the first one?*1374

*Yes I can pick 0.*1377

*Can I pick any score for the 2nd one?*1379

*Sure, I can pick 100.*1383

*Now that third score I cannot take any score.*1386

*If I pick 72 my mean would not be 50.*1392

*If I pick 42 my mean would not be 50.*1394

*If I pick another 0, my mean would not be 50.*1399

*That is the problem and because of that if this is my data set so far I have been free to vary.*1403

*I freely chose this guy but this last one I am locked in.*1410

*I have to choose 50.*1415

*That is the only way I can get a mean of 50.*1417

*That is what we call degrees of freedom.*1420

*This logic is going to become more important later on, but for now what you can think about is *1423

*because we are deriving other summary statistics from our sample we are not completely free to vary.*1429

*We locked ourselves down. *1437

*We pinned ourselves down and built little gates for us at the borders.*1439

*Now you know degrees of freedom and we know as degrees of freedom or n goes up we see more and more normal like distributions.*1445

*I have drawn three distributions here for you.*1460

*Here you might notice that I have used basically the same picture of a curve for all three of these.*1462

*You might think they have all the same distribution.*1469

*Not true because you have to take a look at the way that I have shown you that t down here.*1473

*The way that I have labeled this x axis or t axis in this case is really to change our interpretation of these curves.*1482

*Remember what the normal distribution says.*1493

*The normal distribution says 1 standard deviation to the right or positive side, 1 standard deviation *1496

*to the negative side that area should be about 68% of your entire curve.*1502

*Is it true here?*1507

*No it is not, this does not look like more than 50% of the curve.*1510

*This looks like maybe 1/3.*1521

*Maybe a little less than 1/3.*1526

*This is starting to look more like 60% of the curve, but still maybe not quite 68% of the curve.*1528

*It is still only looks like may be 50% of the curve or a little more.*1539

*Imagine that this was shifted in the middle this would be more like 68% of the curves.*1544

*Something like this would be more like 60% of the curve.*1560

*That is how you can see that as your degrees of freedom increases it becomes more and more normal.*1567

*Even this is not quite normal. *1582

*This is not quite 68% but a little bit less actually.*1585

*As the DF gets bigger and bigger that area starts to look more and more like the normal distribution.*1588

*Now there is another way I can draw these pictures and I believe in this other way you can see more easily how helped this is more variable version.*1598

*Remember I am saying that t distribution is like you are stomping down on the peak of it and smooching it out a little bit.*1615

*I believe that if I draw the same picture in a slightly different way you will see why. *1624

*In this case, here is what I have done. *1630

*I have kept the t axis the same and now it is labeled in the same way, but I have drawn these distributions in a slightly different way. *1634

*Now this one is a little wider and this one is less wide and this one is even less wide.*1647

*It becomes more narrow, more like the normal distribution.*1656

*Notice that if I drew the line here, a little bit after 1 standard deviation away we see they are a little of that curve on the side.*1661

*You know if that is 50% and maybe 15%, 10%, something like that. *1675

*This might look more roughly equivalent to this, maybe a little bit less.*1685

*Maybe like 20%.*1693

*This looks like much more than this.*1695

*Maybe this is like 25 or 30% even compared to this.*1700

*In that way you can see using the same concepts the drawing and picture in a slightly different way that this distribution is much more variable. *1706

*It is spread is very wide. *1719

*Where is this distribution is much less variable?*1721

*Remember t is all because of the variability found in s.*1725

*When s is very, very variable and n is very small, s is very variable, so the t distribution is also quite variable.*1731

*As s n gets bigger, s gets more and more accurate, more like the actual standard deviation of the population.*1741

*And because of that, it becomes more and more normal.*1752

*Let us break this one down.*1755

*In degrees of freedom of 60, here is what it might look like.*1761

*It might look something that is very close to our 34, 13 ½ , 2% normal distribution.*1769

*If we drew our little lines there, that would probably look very close to this picture.*1777

*It looks pretty close.*1792

*When we draw something like this, this area might only be 25% of this whole curve.*1797

*This other areas also combined 25%.*1810

*If I split this like this, then this would be something like 14%. *1817

*A little bit less than this but still quite a bit.*1826

*This one might even be more than 14%, maybe like 18%. *1832

*As you can see that in this distribution even though I have drawn it like this and just labeled it differently. *1840

*In reality, it will look more like this if you kept this t axis to be constant.*1849

*It will look sort of smooched out.*1855

*How do you find t at the end of the day?*1859

*How do you find the t and not only that how do you find the probability associated with that t?*1867

*For instance, where t is greater than 2?*1874

*How do you find these probabilities?*1878

*We know how to do it for z but how do you do it for t?*1881

*One thing that you could do is you can look at the back of your book usually in the appendix section *1884

*there is something called the t distribution or the students t distributions that you can look at.*1892

*Oftentimes it will have degrees of freedom on one side like 2, 3, 4, 5 all the way down and then it will show you either one tailed or two tailed area.*1898

*It might give you .25, .10 and .05, .025.*1914

*It might give you these areas.*1926

*The number right here tells you the t score at that place.*1929

*If you wanted to know where the 25% cut off is, what this t score is for degrees of freedom = 2 distribution and you would look right here.*1935

*If you wanted to know it for .025 then you would look here.*1962

*You want to look for degrees of freedom, as well as how much of the curve you're trying to cover.*1975

*That is definitely one way to do it.*1984

*The other way you could do it is by using Excel and just like how Excel will help you find probabilities *1988

*and z scores for the standardized normal distribution you can also find it in Excel for the t distribution.*1995

*It needs a couple of hints.*2003

*Let us start off with tdist.*2006

*Tdist is the case where you want to find the probability but you have everything else.*2012

*What the tdist will do is if you put in the degrees of freedom and you put in the actual x value.*2019

*You can think of the x value as the t value and it will only take positive t values.*2033

*For instance, a t value of 1 and the number of tails if you want this entire area or you just want that area alone.*2039

*You can either put in one or two then it will give you the probability of this area.*2058

*I can show you right here.*2066

*Let us put in tdist for t(1) and degrees of freedom 2 and let us look at what it might say for two tails.*2070

*It will say 42% and if you look at this exact same thing, but if you look at it for one tail it will just divide this area in half.*2098

*21% and 42% makes sense.*2111

*Basically this is giving you this area + this area if you want 2 tails.*2116

*But if you only want one tail it will just give you this area.*2122

*We know that for 95% competence interval we often use z score of 1.96 and that will give us a tail of .025 or if we count two tails 25%. *2125

*Let us see what this gives for 1.96 when we have a degrees of freedom of only 2.*2149

*Let us put in 1.96.*2158

*If we put that in our z score, if we put in 2 tails we would only get 5%, but let us see what we get here.*2163

*Degrees of freedom 2 and number of tails let us put in 2.*2173

*Do you think this should be more or less than 5%?*2179

*Let us think about this.*2183

*The t distribution is like slightly smooched, it is more spread out and because of that it is going to have this longer tail.*2186

*It is not going to be nice and all compact in the middle.*2196

*It will be spread out.*2200

*We would imagine that it have a fat tail.*2202

*I would say more than 5%.*2204

*We see that it is almost 20% a t of 1.96.*2207

*Let us put that same z score in.*2216

*Normsdist this is whenever we want the probability and put a 1.96.*2218

*Here we get the negative side, so we want 1 - and this gives us just 1 tails.*2227

*I am going to change this to 1 tail, so we could look at it. *2241

*Here on one of our tails, one side of it, it is almost 9 1/2% is still out there.*2245

*But when we use the z score only 2 1/2% are still out there.*2253

*Let us look at the same t distribution for a very high degrees of freedom. *2258

*Let us try 60.*2271

*Even with something like 60 we are starting to get very close to the z distribution, but still this guy is more variable than the z distribution. *2273

*Let us see if we could go even higher. *2287

*Instead of 60 I am going to put in 120.*2289

*Notice we are getting closer but still these are more variable than these.*2294

*Let us go a little less.*2303

*Let us go like 1000 and see what happens there.*2304

*We are getting close but still slightly more variable.*2309

*That is a good principle for us to know.*2316

*The t distribution although it approximates normal, it approximates it from one side.*2318

*Here is the normal distribution standards .02499.*2324

*There it is and it is getting closer and closer to it, but it is approaching it from the high-end.*2329

*These numbers are dropping and getting really close to that, but not quite hitting it.*2336

*Now you know how to get the probabilities but what if you have the probably and you want to find the t score?*2345

*What would you do?*2355

*In this case, you would use the inverse t in for inverse.*2356

*Here you would put in the two tailed probability.*2362

*Let us say we want to know what is the t boundary for if we wanted only 5% in our tails?*2366

*Here is the situation I am talking about for this one.*2374

*We had this distribution and we know we want these to be .025, just like a z distribution.*2378

*We want it to .025 but we want to know what these numbers are here.*2393

*We want to know what these numbers are.*2398

*It depends on your degrees of freedom.*2403

*Let us try degrees of freedom of 2, 60, 120, and 1000.*2405

*Let me label this.*2413

*Here we get the probabilities from t dist and here are the probabilities from standardized normal distribution, or the z distribution.*2424

*We do not want the probabilities we actually want the t boundaries themselves and the z boundaries themselves. *2443

*If we want the z boundary at .025 or at 5%, we would use normsin and we put in our probability.*2460

*I forget if it is one tailed or two tailed.*2472

*Let us try one tailed but we would need two tails.*2474

*We get very close to -1.96.*2477

*We just have to memorize that but that is why this is saying at -1.96 you have about 2 1/2% in that little tail. *2489

*Now what about the t?*2501

*In Excel it is inconsistent because z it gives it to you on the negative side, for the t it only gets 2 for the positive side.*2504

*That is confusing but I often do not memorize that.*2512

*I just try out a couple of things until it spits out the thing I'm looking for.*2515

*You have to understand how these things work so that you could predict what's going on. *2520

*We will use t inverse and we want to know the probability and I believe this is going to be two-tailed.*2527

*.05 and degrees of freedom of 2.*2538

*We get .05 and degrees of freedom just to test whether this is one tailed or two tailed.*2543

*Let me put that in.*2563

*I believe you have to give it two tails.*2565

*You have to put in the two tails probability here so that is .05 and the degrees of freedom 2 and this will give us these boundaries.*2570

*This will only give us the positive boundary, but because it is symmetrical, you automatically know the other side.*2580

*This would give us a boundary of 4.3.*2589

*Remember for the z score this boundary will be 1.96 but for a t distribution with the degree of freedom of 2, this would be 4.3.*2593

*That is quite high because remember it is really spread out.*2604

*You got to go way out far in order to get just that 2%.*2609

*What about this boundary for degrees of freedom of 60?*2612

*What do we get then?*2621

*We get something very close to 1.96 but it is a little bigger than 1.96.*2624

*Remember because the t-distribution is more variable you to go farther out there in order to capture just that small amount of .025%.*2630

*That mean 2.5% or .025.*2641

*If we go to 120 we should expect is that boundary to come closer and closer to 1.96 from the big side, but not quite hit 1.96 or more closely 1.9599.*2646

*We are getting close to that 1.96 number, but still it is a little bit higher.*2671

*Finally we will go buck wild and put in degrees of freedom of 1000 we get something very close to 1.96 but still little the higher than 1.96.*2678

*Those are two different ways that you can find the t, as well as the probability that t is associated with.*2692

*Remember the degrees of freedom and you have to know whether you want two tailed probability or one tailed probability.*2701

*As well as your degrees of freedom. *2714

*That is what you will have to know in order to look things up on a t distribution.*2717

*Let us go on to some examples.*2722

*In each of these situations which distribution do you use, the z or the t?*2729

*Is there a 500 million people on Facebook how many people have fewer friends than Diana, who has 490 friends?*2734

*Assume that the number of friends on Facebook is normally distributed and here they give you the sigma.*2742

*We know that you can use the z distribution here.*2749

*Here the researchers want to compare a given sample of Facebook users average number of friends 25 to the entire population. *2753

*What proportion of sample means will be equal or greater than the mean of this group?*2763

*N = 25, but the mean is 580.*2772

*They have an average of 580 friends.*2779

*Here I definitely would not necessarily use z but I also do not have the standard deviation.*2783

*Maybe this is connected to the previous problem.*2796

*If so, if I assume that they come from the whole population and they give us the information for the whole population here.*2800

*If sigma = 100 then I will use z.*2811

*This one I probably left out some information.*2816

*What about this last one? *2820

*Researchers want to know the 95% competence interval for tagged photos given that a sample of 32 people *2822

*have an average of 185 tagged photos and a standard deviation of 112.*2829

*Here it is very clear, since I know s but I do not know the sigma for tagged photos.*2835

*I only know the sigma for friends, but not for tagged photos.*2844

*In this case, what I would do is use the t distribution because I will probably have to estimate *2848

*the population standard deviation from the sample standard deviation.*2855

*Example 2, what we get is that problem and we just have to solve it. *2860

*There are 500 million people on Facebook but how many people have fewer friends than Diana?*2869

*Here it is good to know that we do not need a sampling distribution of the mean.*2874

*We do not need the SDOM.*2880

*In fact, we are just using the population and Diana.*2882

*We could draw the population and it tells us that the population is normally distributed.*2886

*Number of friends is normally distributed and so the mu = 600 and a standard deviation is 100.*2895

*This little space is 100 so this would be 700.*2914

*Diana has 490 friends so here would be 500.*2920

*It is asking how many people have fewer friends than Diana?*2929

*How many have that?*2937

*It is tricky because this will give us the proportion but it would not give us how many people?*2940

*What we will have to do it multiply that proportion to the 500 million.*2950

*This is all 500,000,000 and that is 100%.*2956

*We will need to know some proportion of them that have friends fewer than Diana, fewer than 490.*2961

*We will have to figure that out and so we will have to multiply 500 million by the percentage. *2974

*Let us get cracking.*2981

*We can figure out the z score for Diana and that would be 490 - 600 and ÷ 100.*2984

*I only need to do standard error if I was using the SDOM but I am using the population standard deviation.*3010

*That is often helpful to draw this.*3016

*Here we have about 100 ÷ 100 = -1.1.*3018

*The z score of -1.1 and I want to know the proportion of people who have friends less than Diana.*3031

*You can look this up on the back of your book, so I would just look up the z score of -1.1 or you could put it into Excel normsdist -1.1.*3046

*I should get about .1357 so that would be .1357.*3065

*That is about 13 ½ % of the population have fewer friends than Diana.*3081

*What I want to do is only get 13% of these entire populations and that would be 500 million × .1357. *3089

*You can do this on a calculator, so that × 500 million = 67.83 million.*3103

*Do not forget to put the million part.*3117

*It is not that you only have 67 people who have fewer friends than Diana.*3124

*That would be our answer right there.*3129

*The researchers want to compare a given sample of Facebook users average number of friends a sample of 25 to the whole population.*3132

*What proportion of sample means will be equal or greater than the mean of this group?*3146

*Here I'm going to assume because there is no other way to this problem.*3159

*I am going to assume that we could use the information from example 2 because we are talking about the same thing, the number of friends.*3165

*We actually know the population.*3173

*The population is approximately normally distributed with the mu of 600 and standard deviation of 100.*3176

*Mu= 600, standard deviation=100 and from this I need to generate an SDOM because *3195

*now we are talking about samples of people not just one person at a time. *3205

*Because of that I need to generate SDOM for n = 25.*3211

*The nice thing is we already know the mu sub x bar = mu that is 600 but we actually also know *3216

*the standard error because standard error is standard deviation ÷√n.*3234

*In this case, it is 100 ÷ √25 =20.*3240

*1 standard error away here is 20. *3246

*This would be 580, 560, and so forth.*3255

*It is asking what proportion of sample means will be equal to or greater than the mean of this group?*3262

*Equal to or greater than means all of these and they are just asking for proportions we do not have to do anything to it once we get the answer. *3271

*Well, it might be nice if we could actually get the z score for this SDOM.*3281

*Here, instead of just putting 580 I would want to find the z score here.*3290

*Here are friends but I want to know it in terms of z score. *3296

*It is actually really easy because it is the z score of -1 and we can actually just use the empirical rule to find this out because we know at the mean, *3303

*at the expected value we know that this is 50% and this is 34%. *3318

*If we add that together, the proportion of sample means greater than or equal to the mean *3327

*of this group that = the proportion where z score is greater than or equal to -1 and that is .84%.*3341

*Final example, researchers want to know the 95% competence interval for tagged photos given that *3357

*a sample of 32 people have an average of 185 tagged photos and a standard deviation of 112.*3366

*Interpret what the CI means.*3375

*Here we do not know anything about the population, but we do know x bar which is 185 *3377

*and we do know the standard deviation of the sample s which is 112.*3386

*We also know n is 32.*3393

*Remember when we talk about competence interval we want to go from the sample to figure out where the population mean be.*3396

*What we do is we assume that we are going to pretend SDOM here and we assume that the *3408

*x bar is going to equal the expected value of this SDOM which is 185.*3420

*From there we could actually estimate the standard error by using s.*3428

*Here mu sub x bar = 185 this is assumed not sigma but x sub x bar is s ÷ √n=112 ÷ √32.*3438

*If you pull up a calculator you could just calculate that out 112 ÷ √32 and get 19.8.*3460

*We know how far the jumps are and because we used s we cannot just find the z score we have to find t score.*3477

*We will have to use t score in order to create a 95% competence interval.*3496

*Although I do not know what the t distribution for the degrees of freedom of 32 – 1.*3504

*I do not know degrees of freedom of 31 t distributions looks like.*3515

*We will have to figure that out.*3520

*What we eventually want is this to be .025.*3524

*These are together a combined two tailed probability of 5% and we will have to use t inverse because we already know the probability.*3531

*We want to go backwards to find the t. *3544

*T inverse and we put in our two-tailed probability .05 and put in our degrees of freedom, which in this case is 31.*3548

*We ask what is the t and it says it is 2.04.*3559

*The t right here at these borders is 2.04 and because it is symmetrical we also know that this one is -2.04.*3564

*In order to find the competence interval we are really looking for these raw values right here.*3577

*In order to get that we add the middle point and add 2.04 standard errors to get out here and we subtract out 2.04 standard errors to get out here.*3587

*The competence interval will be x bar + or - the t score. *3605

*How many jumps multiplied by how big these jumps actually are and that is the score right here multiplied by s(x).*3615

*If we actually put in our numbers that is going to be 185 + or -2.04 × 19.8.*3627

*If you just pull out a calculator we could get 185. *3638

*Make sure to put that = 185 even I forget sometimes +2.04 × 19.8 and remember Excel knows order of operations. *3643

*It will do the multiplication part before it does the addition part,*3660

*The upper limit will be 225.39 and the lower limit will be 144.61.*3664

*I just rounded to the nearest tenth and this would be 225.4 and this would be 144.6. *3683

*We need to interpret what the CI means. *3697

*This means that there is a 95% chance that the population mean will fall in between 144.6 and 225.4 that is the interval.*3705

*That is it for t-distributions. *3721

*Thank you for using www.educator.com.*3724

*Hi and welcome to www.educator.com.*0000

*We are going to be talking about hypothesis testing today.*0002

*The first thing we need to do is situate ourselves where do hypothesis testing fit in with all of inferential statistics.*0005

*We are going to talk about how to create the hypothesis that we are going to test and that hypothesis is going to be about a population.*0015

*When we say about a population we mean about population parameters.*0023

*There is actually two parts to any hypothesis that we test.*0028

*There is the no hypothesis and the alternative hypothesis.*0033

*We are going to talk about how they fit together.*0036

*We are going to talk about potential errors in hypothesis testing because it is good to know going into it.*0039

*Finally, we are going to end with the steps of hypothesis testing and we are going to do the steps of hypothesis testing, *0045

*When sigma the population standard deviation is given and when it is not given.*0051

*And if you had just refresh yourself with the confidence interval lesson, *0057

*You can probably guess that when sigma is given we are going to be using z distributions or normal distributions.*0063

*When sigma is not given and we have to estimate the population standard deviation from the sample using s then we will use t-distributions.*0072

*In order to use the t distribution we need to figure out the degrees of freedom.*0086

*Let us go back and situate ourselves with all of inferential statistics.*0094

*Basically the idea of inferential statistics is that we use some known populations to figure out the sampling distribution.*0101

*The one that we are using a lot is the SDOM.*0115

*We are going to use the another one later.*0121

*We figure out sampling distributions and now we want to compare a sample from an unknown distribution.*0123

*We want to compare sample from that to the sampling distribution. *0136

*If the sampling distribution says the sample is very likely then we might say maybe the sample, *0145

*this unknown population is very similar to the known population.*0154

*But if the sampling distribution tells us the sample was very unlikely then we could rule out *0159

*the known population as a potential candidate for this unknown population.*0169

*In doing all of this in inferential statistics there are two issues that come up.*0176

*What happens when we do not know what the population looks like at all and *0183

*We want to try to figure out where the population mean or different parameters of the population might be.*0188

*In that case we use confidence intervals and when we use confidence intervals we try to figure out where mu is from x bar.*0195

*Another way of thinking about it is we try to figure out something about the population *0211

*From the sample information because we have that sample information. *0216

*Another technique that we could take is that we could use this idea and say how do we decide when a sample is unlikely?*0220

*How do we decide when to draw x?*0235

*When do we decide this side is weird?*0238

*In order to do that we now have to learn about hypothesis testing.*0243

*The goal of hypothesis testing is to be slightly different from confidence interval yet related.*0248

*It is the flip side of the coin. *0254

*Basically, you are going to try to figure out whether your x bar is unlikely given a hypothetical population.*0256

*In that case, what we are doing is we are setting up a population.*0281

*It is like the population is stable and we are going to compare the sample to it.*0290

*Here is our sample and here is our set standard.*0296

*Here the population is moving but this is the target and this is what we use to get that target.*0305

*Here this is already set and we are comparing this guy to this guy.*0316

*In this way you need both confidence intervals and hypothesis testing to give you the full story. *0323

*You might also hear that hypothesis testing another word or phrase for it will be a test of significance.*0331

*A lot of students misinterpret that to be a test of importance.*0343

*That is the modern way the word significance is used but that is not actually what we are talking about here. *0348

*When we call this at test of significance this is actually using the meaning of significance *0354

*from the early 20th century when this test was actually invented.*0367

*Back then significant adjustment prominence or standing out.*0370

*I like to think of it as being weird like how much does this sample stand out?*0377

*Is that significant?*0386

*Is it prominent and different or is it very, very similar?*0387

*Those are the ways you could think about it. *0392

*I do not want to think of it as a test of importance.*0398

*Now that we know why we need hypothesis testing, how do we hypothesize the population?*0401

*How do we make up a population?*0411

*Do we have to make up all the individual numbers of the population?*0413

*What do we got to do?*0415

*Here is the thing, we could assume things about population parameters and test those assumptions. *0417

*We do not have to stimulate every single member of the population we could just make some assumptions about parameters.*0424

*In order to set up a hypothetical population you set up a parameter. *0431

*For instance, you say mu is equal to something.*0437

*That is how you set up a population then check whether our sample is likely to have come from such a population.*0440

*In doing this we need to figure out how to we hypothesize rigorously so that we could get as much paying for our book from our hypothesis?*0448

*In order to do this we have two parts to a hypothesis and this is going to make our hypothesis better.*0462

*The first part of hypothesis is what we call the null hypothesis and null means 0 or not important.*0472

*The null hypothesis in this case is your hypothetical population.*0487

*We write the null hypothesis like this h sub 0 or h sub knot.*0492

*We might say mu= 0.*0502

*We have created a null hypothesis. *0507

*I just made up to 0 but there are better ways of doing this and we will talk about those later.*0510

*We could also write this in terms of standard deviation or other things but frequently *0516

*you will see the mean being the hypothesis of the population.*0532

*The alternative hypothesis is what do we learn if this is not true?*0536

*If we rule this out then what have we learned?*0544

*In that way these two make up the full hypothesis. *0548

*If we find this then we learn this.*0554

*If we do not find that we learn this other thing.*0557

*What we learn if this is not true is at least that mu does not equal 0.*0560

*This is called the alternative hypothesis and it helps us at least figure out something when we do not figure out that.*0566

*If we do not find this to be true at least we find this to be true.*0575

*If this is not true then we will always find this to be true.*0580

*These two hypotheses together this is more powerful than just having one hypothesis alone.*0584

*We will talk a little bit about why and it goes back to that idea of the test of significance.*0597

*Hypothesis testing or the test of significance is a test of weirdness.*0607

*It tests how weird the x bar is.*0617

*This is the question that it can answer is the x bar weird?*0625

*Is it different from the population?*0633

*But it actually tell is x bar very similar to the population?*0636

*That is not what number gives you but only tells you how weird it is.*0642

*It does not tell you how similar it is.*0646

*These are actually not flip sides of the same coin and because of that our goal here in all *0648

*of hypothesis testing is we find out the most when we reject the null hypotheses.*0658

*That is when we would find out the most.*0668

*This may not seem like we are finding out of luck because we ruled out 0.*0671

*There is an infinite number of mu that we need to test but actually in hypothesis testing *0676

*what you want to do is reject the no rather than accept or fail to reject the null.*0682

*Just because it is set up as a test of weirdness that is the only thing you can find out.*0690

*It is true that it would be nice if we can find out more than that but that is the limitation of this hypothesis testing.*0695

*It is a limitation that is also like the fact of life because even as the limitations this hypothesis testing still a powerful tool.*0703

*But it is good to keep in mind that this one is a limitation.*0714

*A little bit more about these two hypotheses. *0716

*These two hypotheses, the null and the alternative, sometimes you might see the alternative written as h sub 1.*0722

*They must be mutually exclusive. *0729

*This means if one is true the other cannot be true.*0732

*If the other is true, the first cannot be true.*0736

*You cannot have a null hypotheses and alternative hypotheses like mu=1 and mu=2.*0739

*They are not mutually exclusive.*0748

*If one is false, the other one does not have to be true.*0751

*It could be true but it does not have to be.*0755

*Whereas mu does not equal 1, mu = 1.*0758

*Those are mutually exclusive. *0763

*If you rule out one you absolutely know that the other one has to be true.*0764

*Together they must include all possible values of the parameter.*0768

*You can think of the parameters such as mu on a number line and you need to cover the entire number line.*0774

*You can have a null hypothesis like mu > 0.*0782

*You might say mu >0 but then your alternative hypotheses have to be mu < or = 0.*0788

*You color that in and color all of that in too because that is where you will cover the entire space, the parameter space.*0800

*If these are both true, here is what you get.*0811

*One of these two hypotheses must represent the true condition of the population.*0815

*You find out something that is true about the population and then as we said before, *0821

*typically in research your goal is to reject the null and find support for the alternative hypothesis. *0827

*You can actually prove the null hypothesis but you can reject the null hypotheses.*0833

*And the whole reason is because hypothesis testing is a test of significance or test of weirdness.*0838

*This x bar stands out.*0848

*You can only tell me whether it stands out a lot from the population or not.*0851

*They can tell me it is probably similar to the population.*0856

*You cannot tell me that part.*0860

*Let us talk about some errors that we could potentially make in hypothesis testing. *0862

*There are some foibles, you need to watch out for.*0868

*Well first, it helps to imagine that there are two potential realities and we do not know which one of them is true.*0871

*One is that the null hypothesis is true.*0883

*It is actually true.*0887

*We do not know yet, but it is true.*0888

*Other possible reality is that the null hypothesis is false.*0892

*Your sample did not come from the population.*0898

*Those are your two possible realities but only one can be true at any given time. *0901

*You cannot have both the null population being true and false at the same time.*0907

*You got to have one or the other.*0915

*These two boxes, this one and this one have to add up to 100%, but these two boxes , this one and this one have to add up to 1.*0916

*That is because we have a 100% possibility of this being true and 100% possibility of this being true.*0934

*If this is true then this is not true.*0942

*Given that this is reality but we do not know reality, what is the deal?*0944

*How do we put that together with hypothesis testing?*0955

*When we do have hypothesis testing we have 1 of 2 outcomes. *0959

*We could either reject the null successfully, that is what we wanted to do.*0964

*We could either reject the null or we can fail to reject the null.*0968

*We do not call this accept the alternative or accepting the null.*0972

*We call it failing to reject because that is how much we wish we could have rejected the null.*0980

*We failed to reject the null. *0987

*Let us think about these two decisions in conjunction with reality. *0989

*Here is the thing, when we reject the null hypothesis and say this sample did not come from the population. *0997

*If it did not come from that population we would be correct here. *1006

*This would be a correct decision. *1011

*If this is our decision and this is indeed the world we live in, this is a correct decision.*1014

*If we fail to reject the null however but the null is actually true we should not have rejected it *1021

*then this also represents a correct decision.*1034

*Good job not rejecting the null because it is right all along.*1039

*These two are ways that we could be correct.*1044

*That leaves us two ways that we could be incorrect. *1048

*One way is this, we could successfully reject the null but the null is actually true but *1051

*we said that it is false but the null is actually true. *1063

*This is an incorrect decision.*1068

*We call this a false alarm because we are rejecting that now.*1074

*It is false alarm we should have not rejected that null. *1084

*The probability of that false alarm is represented by the term alpha.*1088

*On the other hand, there is another way that we could be wrong and that way is this.*1097

*We could fail to reject the null.*1107

*We could say we may not be wrong. *1109

*We fail to reject it but the null is wrong.*1114

*This is also an incorrect decision.*1121

*This is not called a false alarm instead it is called a miss.*1127

*This is going to be called the beta rate.*1134

*Obviously the alpha and the beta have a probability of less than 1, but greater than 0.*1143

*What we want to do in hypothesis testing is reduce our chance of errors.*1150

*We can also figure out what is our probability of getting different kinds of correct decisions?*1157

*We know that this is one version of the world and that should add up to 100% this probability of failing to reject when we should have kept it around.*1167

*This probability is 1 – alpha.*1183

*This is what we call a correct failure.*1188

*It sounds odd but it sounds good that you have failed.*1198

*You failed to reject it and you should have failed to reject it.*1203

*It is like you failed to reject a date and you know that date was really good.*1208

*He is a good guy so you should have failed to reject him.*1216

*On the other hand, this is another possible set of what could be right in the world.*1225

*This should add up to 100%, so this should be 1 – beta.*1232

*That is our rate of correct decision where we successfully rejected the null and it is indeed false.*1238

*In dating it might be reject somebody who comes up to you and good job you should have rejected them.*1245

*They are a total loser.*1253

*That is what we call a hit.*1255

*It is like in a battleship when you hit it.*1258

*This is the hit rate, miss rate, false alarm rate, and the correct failure rate.*1263

*Let us talk about the steps of hypothesis testing. *1272

*Well there are going to be 5 steps.*1281

*The first step just starts out with setting up your hypothetical population.*1284

*This is the hypothetical population and you need to create both a null hypothesis and an alternative hypothesis then pick a significance level.*1290

*You can think of the word significant as a stand outness like how much it standout.*1304

*How much does it have to standout?*1310

*When it stands out a lot you have a very low false alarm rate.*1313

*If your x bar is out there and then you have a small chance of false alarming.*1318

*You are saying this really does not look like it belongs in the population because it is so out here.*1326

*And that is where your false alarm rate is low. *1335

*You want to set a low one. *1338

*If you want to be more conservative, you want to set an even lower false alarm rate. *1340

*For instance, alpha = .01 that would be even lower rate of false alarm.*1344

*Then you want to set a decision stage.*1351

*So far, we have not done anything except like setting things up yet and still we are setting things up.*1355

*We set up the decision stage and what you want to do is draw the SDOM, the sampling distribution.*1361

*We have the hypothetical population and we create a sampling distribution so that we can take our sample *1368

*and compare it to that sampling distribution. *1375

*You draw the SDOM and you identify the critical limits.*1378

*Here is my SDOM and you want to identify the extreme regions where you say if your x bar *1383

*is somewhere out here then you want to reject the null.*1396

*You want to say it is very, very unlikely to have come from this null population.*1402

*Then choose a test statistic because the test statistic will tell you how far out from the mean it is in terms of standard error.*1407

*How many jumps out you are?*1419

*This will be called choose a critical test statistics.*1421

*You are saying what are the extreme boundaries such that if x is outside those boundaries we reject it.*1429

*If it is inside the boundaries we do not reject.*1440

*And then we use the sample. *1444

*This is the first time we are doing anything with the sample.*1447

*We use the sample and the SDOM from here to compute the sample test statistic and p value.*1450

*And the p value is going to tell you given that x is out here how much of that curve does it actually cover?*1458

*What is the probability of false alarming there at that particular value?*1468

*And then you compare the sample to this SDOM population and you decide to reject the null or not?*1476

*One word about p value versus alpha.*1487

*The p value is going to be the probability of belonging to the null population given sample x bar.*1494

*What is the probability that this value belongs in here?*1513

*Alpha is what we call the critical limit. *1519

*This is what we are able to tolerate we just set it.*1526

*Alpha is often decided just by the scientific community. *1532

*In fact alpha is often set to something like .05 or .01 because that is commonly accepted in scientific communities.*1536

*We call that just being by tradition or convention.*1546

*It is not that we figured out the alpha level.*1550

*On the other hand we figure out the p value level given our sample x.*1553

*And what we want is for the p value to be lower than the critical limit.*1559

*Let us go through some examples.*1566

*Here is an example of single sample hypothesis testing, also called t tests of 1 mean or single mean t test.*1572

*This is also another term for it.*1594

*Let us talk about this when sigma is available. *1597

*The population standard deviation has been given to us.*1601

*Here it says that the average Www.www.facebook.com.com user has 230 friends, a sigma of 950, a random sample of college students n=39 showed that the sample mean was 393 friends.*1605

*Our college students like the average www.www.facebook.com.com user.*1620

*Let us try to think about this by using hypothesis testing.*1624

*The first thing is perhaps we should set up the best standard population as the average www.www.facebook.com.com user,*1631

*the real population of all Www.www.facebook.com.com users.*1643

*Our null hypothesis might be something like mu= 230.*1648

*That the null hypothesis is that our college students sample is just like everybody else. *1655

*The alternative hypothesis is that our samples are not similar to that population. *1667

*Let us set the significance level. *1678

*Here we could just use alpha = .05 by convention.*1683

*We could say that is traditional, we will use that too.*1693

*Let us set the decision stage. *1698

*Here we want to start off by drawing the SDOM and I like to label for myself that it is the SDOM *1701

*just so that I do not get confused and mistake it for the population or something like that.*1711

*We want to draw a critical limit.*1717

*If this is the only false alarm that we are willing to tolerate then we might say everything out here we reject.*1721

*Everything out here we reject.*1730

*That would mean that everything in here is 95% and out here these two regions together add up to 5%.*1734

*Because we are going to reject it there is still some probability that this sample belongs to the population.*1745

*But we are going to reject the null.*1751

*We need to split up 5% distributed to both sides so this would make this 2.5% and this would be also 2.5%.*1754

*That is the error that we are going to tolerate.*1768

*I will color n right now my rejection regions so that means if it out here in the extremes I am going to reject my null hypothesis.*1771

*And because we know that this SDOM comes from the population, that is how we are creating this SDOM.*1783

*We know that the mu of SDOM is exactly equal to the mu of the population so that will be 230.*1792

*Mu sub x bar = 230.*1801

*We can also figure out the sigma sub x bar and that would be just sigma ÷ √n Which is 950 ÷ √239.*1805

*You could just pull out a calculator to do this.*1819

*I am just going to use the blank Excel file and here is 950 ÷ √239= 61.5.*1823

*That is my standard error of this population.*1839

*And what I want to know is it is nice to have that but if it would also be nice to know what is the z score out here?*1848

*We use z score because we are using sigma.*1856

*What is the z score out here?*1861

*Actually I had just made you memorize it when we previously talked about confidence intervals so we know that is 1.96 and -1.96.*1864

*If you wanted to you could also figure it out by using either the table in the back of your book or Excel *1876

*so we could put in normsin because we have the probability.*1885

* I want the two tailed probability this is actually one tailed.*1890

*The one tailed probability is going to be .025 way down here.*1902

*This little bottom part down here it is covered .025 of this and Excel is telling me that the z score right there is about 1.96.*1910

*Now that we have all of that settled, we could start tinkering with our actual sample. *1924

*Let me draw some space here.*1933

*Let us talk about our sample.*1938

*When we talk about our sample we should figure out how far away is our sample mean?*1942

*We just do not want to know in terms of how far away they are in terms of friends but we want to know *1955

*how far away in terms of the standard deviation because only standard deviation will tell us what proportion of the curve is colored.*1962

*Even if we find out the actual raw distance away 163, we do not know where that is in relation to this curve.*1971

*It would be nice if we could find the z score of 393 then we will know where it is in relation to this curve.*1983

*That would be 393 – 230 so how far is it away from 230, all divided by the standard error 61.5 *1990

*because that will give me how many standard errors away we are.*2002

*Let me just calculate that.*2007

*That would be 393 - 230 and I need parentheses because I need it to do the subtraction before the division and that gives me 2.65. *2011

*My z score is 2.65.*2032

*Here this maybe 1 z score away, this is almost 2 z scores away and let us say this is 3 z scores away. *2036

*I know that my 393 is somewhere around here because it is around 2.65.*2049

*This area is very tiny, so I need to find the p value here. *2061

*What is the p value here?*2070

*What is the probability that x bar is greater than or equal to 393?*2072

*That equals the probability that z is greater than or equal to 2.65.*2091

*Not only that but remember we have a two tailed hypothesis.*2100

*We are interested in either being greater than or less than the mean.*2106

*We actually have to find this thing out and multiply it by 2.*2112

*What you can do is look this up in the back of your book and multiply it by 2 or Excel will actually calculate it for you *2117

*like you could put in normsdist and put in the negative side because normsdist gives it to me going from the negative side to positive side.*2128

*I am going to color this part first.*2143

* -2.65 and it should be a very tiny number that will be .004.*2144

*That is a tiny number and then we take that one side and we multiply it by 2 to give us our p value.*2153

*What we are really doing is we are coloring this base, pretend that is inside and also getting -2.65 *2160

*and coloring that space and adding those two together.*2179

*That will give us .008.*2183

*What about a single sample hypothesis test when sigma is not available?*2188

*Well this is the exact same problem in fact I have crossed this out so you can no longer use it. *2201

*It is no longer available to you.*2208

*Here what we have to do is estimate sigma and use s instead of sigma.*2212

*Let us go ahead and start off just hypothesis testing.*2219

*Our null hypothesis is mu=230 that are our sample of college students is just like everybody else. *2222

*Our alternative is that they are different from everybody else. *2233

*Different in some way, either have more friends or less friends.*2239

*We also need to pick a significance level.*2244

*How extreme does this x bar have to be?*2248

*We are going to pick alpha=.05 just by convention we do not figure it out or anything. *2255

*And then we need to set our decision stage. *2260

*Here we want to start off by drawing our SDOM helps to keep this in mind that this is a bunch of means, a bunch of x bars.*2264

*We can just use this information because this is our known population.*2276

*We are going to use that information to figure out our SDOM.*2284

*Here we run into the problem how can we figure out standard error?*2288

*Well, we cannot figure out sigma sub x bar but we can actually figure out s sub x bar.*2294

*That standard error using s instead of sigma.*2302

*That will be s(x) ÷ √n. *2307

*We have s for more sample, the standard deviation of our sample which is 447 ÷ v239.*2316

*And I will just pull out my Excel in order to calculate this.*2326

*447 ÷ v239 and I get 28.9.*2346

*I am actually going to draw in my rejection regions, anything more extreme is going to be rejected.*2356

*Fail to reject in the middle and this rejection region is .025 and this rejection region is .025 because *2375

*I need to split that significance level in 2.*2389

*What we do here is we want to figure out what is our actual t statistic?*2393

*How many standard errors we are when we talk about these borders?*2404

*What is our critical t?*2408

*That would be the t values here.*2410

*This is our raw values in terms of friends but we want to know it in terms of standard error.*2413

*Here are our t values so we cannot just put in 1.96 because that would be for z distributions.*2418

*We need a t distribution and in order to find a t distribution we need degrees of freedom.*2426

*The degrees of freedom is n-1 and that is 238 because 239 – 1.*2434

*You can either look this up in the back of your book or I am going to look this up on Excel.*2443

*Here I am going to use my t inverse and I put in my two tailed probability .05 and my degrees of freedom which is 238.*2451

*And I get 1.97.*2465

*1.97 and -1.97 because t distributions has many problems as they have they are perfectly symmetrical.*2470

*Those are critical t.*2485

*That is the boundary t values.*2488

*Now we have all of that, now we can start thinking about our sample. *2491

*Let us think about our samples t and p value.*2499

*The sample t would be the distance that our sample is away from our mean ÷ standard error because we want how many standard errors away we are.*2505

*393 - 230 ÷ standard error 28.9.*2523

*I will put that into my Excel 393 – 230 ÷ 28.9 = 5.6.*2532

*Let us find the p value there.*2546

*We know that it is far out here our t value so this is about 2, 4, 5.6.*2552

*It is way out here.*2560

*Imagine this going all the way out here.*2562

*That is where x bar landed.*2565

*Already we know that it is pretty far out but let us find the precise p value there.*2569

*In order to find the p value we want to use t dist because that is going to give us the probability.*2577

*We put in the x and that is Excel's word for t.*2583

*When you see x here in t distribution just put in your t value and it only accepts positive t values.*2588

*I will just point to this one, our degrees of freedom which is 238 and how many tails?*2600

*We have a two tailed hypothesis.*2609

*We get 4.8 × 10*^{-8} so that would be our p value.2612

*Our probability of getting a t that is greater than or equal to 5.64 or t is less than or equal to -5.64 because it is two tailed equals 4.8 × 10*^{-8}.2624

*Imagine .07 × 48 and so that is the pretty number.*2658

*This number is so small that they cannot even show you the decimal places.*2669

*It is super close to there but not 0.*2677

*This is our p value, is the p value less than .05?*2680

*Indeed it is.*2686

*What do we do?*2688

*We reject the null hypothesis.*2691

*This is what we do when sigma is not available.*2695

*Just to recap about alpha versus p value. *2702

*P value is the probability of seeing that sample t or an even more extreme statistic given that the null hypothesis is true.*2709

*And we say extreme because they can be like way bigger or ways smaller either side right.*2720

*Alpha gives you the level of significance. *2729

*That level of extremeness that you have to reach in order to reject your null.*2733

*This is the set standard.*2739

*And this is the thing that you are going to compare to that set standard. *2742

*I want to talk briefly about one versus two-tailed hypotheses.*2751

*When we talk about a one tailed hypothesis, you might have something like mu is going to be greater than 0.*2757

*Or your alternative will be mu is less than 0.*2768

*If that is the case and your set alpha level is .05 then here is what you would do in your SDOM.*2777

*You will only use one side of it because you are not interested if your x values are way up here. *2786

*You only care if your x value is way smaller than your population. *2798

*In this case, you might set up this as your rejections zone and notice that it only on one side because one tailed and these are end tails.*2805

*That probability will be .05 and this failed to reject side will be .95.*2817

*This is a one tailed hypothesis. *2830

*Frequently we will be dealing with two tailed hypotheses.*2833

*In that case that might be that you do not really care. *2838

*We do not really care if mu is less than, way smaller or way bigger than what we expected. *2845

*We just care if it is extreme in some way, different in some way.*2854

*We do not really care which way and that would be mu = 0 and the alternatives is that mu do not = 0.*2858

*If we had something like alpha = .05 in a two-tailed hypotheses then we would split up *2868

*that rejection region into the two-tails so that will be .025 and .025. *2879

*We reject , we reject, but inside of these boundaries we fail to reject and this is 90.95%.*2889

*Whatever p value you find we want to compare it to the set alpha level.*2906

*Let us talk about some examples.*2915

*Your chemistry text book says that if you dissolve table salt and water the freezing point will be lower than it is for pure water 32°f.*2920

*To test this theory, your school does an experiment with 15 teams of students dissolved salt and water and put them in the freezer with the digital thermometer.*2931

*Periodically checking to observe the temperature at which the solution freezes.*2940

*The data is shown in the download below. *2945

*What can you conclude from this data?*2948

*If you look at your download and go to example 1, here are all my freezing temperatures that each of my teams got *2951

*and I think there are only 14 teams here.*2963

*Let us suggest that to be 14. *2967

*What should we do first?*2969

*Just to give you an example of what it is like to do one tailed hypothesis testing, let us have a one tailed test here.*2973

*Because it does say that putting the salt and water the freezing point should be lower *2982

*that automatically gives us a direction that we expect, the freezing point to go in.*2990

*What would our null or default hypothesis be?*2999

*The default hypothesis would be that it is not different from pure water. *3004

*They are the same.*3010

*It might be something like mu=32°f.*3011

*But do we care if our samples are all greater than 32°?*3019

*Maybe the freezing point is higher.*3028

*Do we really care about that?*3032

*No not really. *3035

*Null hypothesis is really that we do not care if it is anything higher than or equal to 32°.*3037

*What we eventually want to know is it lower like weird in this low direction.*3051

*The alternative hypothesis is that it is weird, but in a particular direction that it is too low way lower than 32°.*3058

*Our Alpha is going to be .05, but let us make it clear that it is one tailed. *3071

*Usually they do not say anything but most people assume two tails as the default.*3079

*Let us say one tailed.*3086

*Let us draw this SDOM for the decision stage and here is idea. *3088

*The default is that all the samples come from a population with 32° is the mean of this SDOM but *3096

*we want to know is it weird and a lot lower than that?*3113

*It is consistently lower than that.*3126

*That is our rejection region and that rejection region is going to be .05 because our fail to reject region is going to be .95.*3128

*Now that we have that it would be useful to know what our t statistic here.*3144

*This is raw in terms of degrees Fahrenheit.*3150

*We also want to know the t statistic.*3156

*Here at 0 what is the t statistics here that looks like boundary?*3159

* In order to know that we need to figure out a couple of things.*3164

*I will start with step 3, one of the things I want to know is that t statistics there.*3168

*In order to find that t statistics we need to know degrees of freedom for the sample and that is just account how many axis we have in our sample -1?*3179

*That is 13° of freedom. *3193

*What is the t value there?*3196

*We have the probabilities and we want to know the critical t or boundary t.*3199

*In order to know that we need to use t in here it asks for a two tailed probability.*3212

*We need a one tailed hypothesis so we have to turn that into a two tail probability. *3221

*If this was a two-tailed it would it be .1 and the degrees of freedom is 13.*3228

*It will only give you the positive side, but we could just turn it into -1 because it is perfectly symmetrical.*3237

*This critical t is -1.77.*3248

*Okay, now that we have that, we can start on step 4.*3252

*Step 4 deals with the sample t.*3259

*In order to find the sample t we probably need to find the mean of sample and that is average and we probably also need to know the standard error.*3264

*In order to find standard error what we need is s ÷ √n.*3289

*It is not like for Excel, this is just for me as I need to know s.*3299

*What is my s?*3304

*That would just be stdv in all of these.*3308

*Once I have that then I could calculate standard error s ÷ √n Which is 14.*3314

*We have a standard error, we have a mean, now we can find our sample t *3327

*and that is going to be the mean of the sample - the hypothesized mu 32 ÷ the standard error.*3334

*I get -3.7645.*3347

*We know that this is much more extreme on the negative side than -1.77. *3354

*We also need to find the p value. *3363

*What is the p value there?*3366

*We need to use pdist because we do not know the probability there.*3370

*We put in our t value but remember Excel only accept positive one and I am only going to put so two – is +.*3376

*The degrees of freedom, which is 13 up here and how many tails?*3390

*Just one.*3398

*That is going to be .001 p value.*3399

*Since I have ran out of room I will just write the p value here so p = .001.*3407

*Is that p value smaller than this alpha?*3416

*Yes, indeed. *3420

*What can we say?*3421

*We can reject the null.*3424

*What can I conclude from this data?*3426

*I can say that this data shows that it is very unlikely to come from the same population as pure water.*3430

*The freezing point of water will have a variation.*3445

*It will have some probability of not being exactly 32 and this deviation on the negative side is much greater than would be expected by chance.*3449

*Let us see.*3461

*Example 2, the heights of women in the United States are approximately normally distributed with a mean of 64.8 in.*3465

*The heights of 11 players on a recent roster of the WNBA team are these in inches.*3472

*Is there sufficient evidence to say that this sample is so much taller than the population that *3479

*this difference cannot reasonably be attributed to chance alone?*3485

*Let us do some hypothesis testing.*3489

*Here our null hypothesis is that our sample is just like regular women.*3493

*The mean is 64.8. *3500

*I am going to use a two tailed alternative here, is that they are not like this population.*3504

*We can probably guess by using common sense that they are on average taller, but we will do a two-tailed test.*3514

*It is actually more conservative. *3522

*It is safer to go with that two tailed test.*3525

*Here we will make alpha=.05 and it will be two-tailed.*3527

*Let us draw the SDOM here.*3536

*Here we might draw these boundaries and because it is two tailed this is .025 .025 and here it is .95.*3542

*All together it adds up to .1. *3565

*Now that we have this can we figure out the t?*3568

*In order to figure out the t, we need to have the degrees of freedom. *3575

*If you go to the download and go to example 2, I have listed this data here for you and we can actually find the degrees of freedom here.*3579

*Here I put step 3 so that we know where we are.*3590

*In step 3, we need degrees of freedom and that would be count of all of these guys -1.*3596

*We have 11 players 10° of freedom.*3606

*Let us find the critical t. *3610

*The critical t would be t inverse because we know the two tailed probability .05 and the degrees of freedom.*3613

*That gives us the positive critical t.*3626

*That is 2.23 and -2.23 those are our critical boundaries and anything outside of that, we reject the null. *3629

*Let us go to step 4.*3640

*In step 4 we can start dealing with the sample. *3643

*Let us figure out the sample t in order to do that we need the x bar - the mu ÷ standard error.*3646

*We need to know the samples average x bar. *3656

*We also need to know mu and we also need to know standard error.*3663

*Standard errors is going to be s ÷ √n.*3669

*I need to write these things down because it helps me figure out what we need.*3674

*It is like a shopping list.*3679

*Here I need s.*3680

*Now that I have written all these things down I can just calculate them.*3684

*I need the average and mu which I already know from the problem 64.8.*3688

*I need to get my standard error but before I do that I need to get s standard deviation *3709

*and 1 standard deviation I can take that and ÷ the square root of n which is 11.*3718

*That is my standard error and once I have all of these ingredients, I can assemble my t which is x bar – mu ÷ standard error.*3730

*I get 7.97 and that is way higher than 2.2.*3746

*I am pretty sure I can step 5, reject the null.*3755

*If I go back to my problem, then let me see is there is sufficient evidence to say that this sample is so much taller than the population, *3763

*that this difference cannot be reasonably attributed to chance alone. *3776

*I should say yes because when you are way out here, your probability that you belong to this chance distribution is small *3780

*that it is reasonable for us to say that the sample came from a different population.*3793

*Final example, select the best way to complete the sentence.*3802

*The probability that the null hypothesis is true, that is a false alarm rate.*3810

*It is when the null hypothesis is true, but also it is not just that.*3824

*It is not just the possibility that the null hypothesis is true it is that given that you have a particular sample it seems to leave some information.*3835

*It is not quite complete, but it is not entirely false. *3850

*It is just that it does not have the whole truth.*3856

*It does not have the condition.*3859

*Given that you have this particular sample value, the probability that the null hypothesis is false, that is not true.*3861

*Even if you just remember this.*3870

*Remember this column was null is true.*3873

*Alpha is the set one but the p ones are the ones in there.*3877

*That is just not true.*3885

*The probability that an alternative hypothesis is true.*3889

*Actually, we have not talked about that at all.*3895

*We only talked about having a very low possibility that the null hypothesis is true, *3898

*but we have not talked about increasing the probability that the alternative hypothesis is true.*3905

*Beside why would you reject the null when you have a really small t value?*3910

*A small possibility that the alternative hypothesis is true that does not make sense.*3915

*What about the probability of seeing a sample t as extreme as the one given that the null hypothesis is true. *3921

*This is our entire story I can process it now.*3934

*It is not just that the null hypothesis is true, but it also that when you have a certain sample, that also has to be part of the definition of p value. *3938

*The idea is if we have this t value and it is pretty extreme and the null hypothesis is true.*3956

*That is given.*3967

*Given that the null hypothesis is true, what is the possibility of seeing such extreme t value?*3968

*It is very small.*3979

*We are trying to lower our false alarm rate.*3981

*That is the end of one sample hypothesis testing.*3986

*Hi and welcome to www.educator.com.*0000

*Today we are going to talk about confidence intervals for the difference of two independent means.*0002

*It is pretty important that there are for independent means because later we are going to go to non-independent or error means.*0007

*We have been talking about how to find confidence intervals and hypothesis testing for one mean.*0013

*We are going to talk about what that means for how we go about doing that for two means.*0023

*We are going to talk about what two means means?*0029

*We are going to talk a little bit about mu notation and we are going to talk about sampling distribution of the difference between two means.*0032

*I am going to shorten this, this is just means this is not like official or anything as SDOD *0041

*because it is long to say assembling distribution of the difference between two means, but that is what I mean.*0048

*We will talk about the rules of the SDOD and those are going to be very similar to the CLT (the central limit theorem) with just a few differences.*0055

*Finally, we all set it all up so that we can find and interpret the confidence interval.*0066

*One mean versus two means.*0075

*So far we have only looked at how to compare one mean against some population, but that is not usually how scientific studies go.*0081

*Most scientific studies involve comparisons.*0091

*Comparisons either between different kinds of water samples or language acquisition for babies versus babies who did not.*0093

*Scores from the control group versus the experimental group.*0102

*In science we are often comparing two different sets of the two different samples.*0106

*Two means really means two samples.*0112

*Here in the one mean scenarios we have one sample and we compare that to an idea in hypothesis testing *0120

*or we use that one sample in order to derive the potential population means.*0132

*But now we are going to be using two different means.*0140

*What do we do with those two means?*0143

*Do we just do the one sample thing two times or is there a different way?*0145

*Actually, there is different and more efficient way to go about this.*0152

*Two means is a different story.*0155

*They are related but different story.*0159

*In order to talk about two means and two samples, we have to talk about some new notation.*0162

*This is totally arbitrary that we use x and y.*0170

*You could use j and k or m and n, whatever you want.*0176

*X and y is the generic variables that we use.*0182

*Feel free to use your favorite letters. *0189

*One sample will just be called x and all of its members in the sample will be x sub 1, x sub 2, x sub 3.*0191

*When we say x sub I, we are talking about all of these little guys.*0203

*The other sample we do not just call it x as well because we will get confused. *0208

*We cannot call it x2 because x sub 2 has a meaning.*0216

*What we call it is y.*0221

*Y sub i now means all of these guys.*0224

*We could keep them separate.*0229

*In fact this x and y is going to follow us from here on out.*0232

*For instance when we talk about the mean of x we call it the x bar.*0236

*What would be the mean of y?*0241

*Maybe y bar right. *0243

*That makes sense.*0246

*And if you call this b, this will be b bar.*0247

*It just follows you. *0253

*When we are talking about the difference between two means we are always talking about this difference. *0256

*That is going to be x bar - y bar. *0264

*Now you could also do y bar - x bar, it does not matter.*0267

*But definitely mean by the difference between two means.*0271

*We could talk about the standard error of all whole bunch of x bars, standard error of x, standard error of y.*0274

*You could also talk about the variance of x and the variance of y.*0285

*You can have all kinds of thing they need something to denote that they are little different.*0292

*That standard error of x sort and another way you could write it is that we are not just talking about standard error.*0298

*When we say standard error, you need to keep in mind if we double-click on it that means the standard deviation of a whole bunch of means.*0312

*Standard deviation of a whole bunch of x bars.*0322

*Sometimes we do not have sigma so we cannot get this value.*0328

*We might have to estimate sigma from s and that would be s sub x bar.*0334

*If we wanted to know how to get this that would just be s sub x.*0345

*Notice that is different from this, but this is the standard error and this is the actual standard deviation of your sample ÷ √n.*0353

*Not just n the n of your sample x.*0367

*In this way we could perfectly denote that we are talking about the standard error of the x, the standard deviation of the x, and the n(x).*0372

*You could do the same thing with y.*0387

*The standard error of y, if you had sigma, you can just call it sigma sub y bar because it is the standard deviation of a whole bunch of y bars.*0390

*Or if you do not have sigma you could estimate sigma and use s sub y bar.*0402

*Instead of just getting the standard deviation of x we would get the standard deviation of y and divide that by √n Sub y.*0411

*It makes everything a little more complicated because now I have to write sub x and sub y after everything.*0423

*But it is not hard because the formula if you look remains exactly the same.*0430

*The only thing that is different now is that we just add a little pointer to say we are talking *0438

*about the standard deviation of our x sample or standard deviation of our y sample.*0446

*Even this looks a little more complicated, deep down at the heart of the structure it is still the standard error equals standard deviation of the sample ÷√n.*0452

*Let us talk about what this means, the sampling distribution of the difference between two means. *0466

*Let us first start with the population level.*0477

*When we talk about the population right now we do not know anything about the population.*0480

*We do not know if it is uniform, the mean, standard deviation.*0491

*Let us call this one x and this one y.*0500

*From this x population and this y population we are going to draw out samples and *0507

*create the sampling distribution and that is the SDOM (the sampling distribution of the mean).*0514

*Here is a whole bunch of x bars and here is a whole bunch of y bars.*0522

*Thanks to the central limit theorem if we have big enough n and all that stuff then we know that we could assume normality.*0530

*Here we know a little bit more than we know about the population.*0540

*We know that in the SDOM, the standard error, I will write s from here because *0545

*we are basically going to assume real life examples when we do not have the population standard deviation.*0557

*The only time we get that is like in problems given to you in statistics textbook.*0565

*We will call it s sub x bar and that can be the standard deviation of x/√n sub x.*0570

*We know those things and we also know the standard error of y and that is going to be the standard deviation of y ÷ √n sub y.*0585

*Because of that you do not write s sub y again because that would not make sense that *0601

*the standard error would equal the standard error over into something else.*0607

*That would not quite make sense. *0612

*You want to make sure that you keep this s special and different because standard error *0614

*is talking about entirely different idea than the standard deviation.*0621

*Now that we have two SDOM if we just decided to do this then we would not need to know anything new about creating a confidence interval of two means.*0625

*You what just create two separate confidence intervals like you consider that x bar, *0638

*consider that y bar, construct a 95% confidence interval for both of these guys.*0644

*You are done.*0649

*Actually what we want is not a sampling distribution of two means and get two sampling distributions.*0650

*We would like one sampling distribution of the difference between two means.*0661

*That is what I am going to call SDOD.*0668

*Here is what you have to imagine, in order to get the SDOM what we had to do is go to the population and draw out samples of size n and plot the means.*0671

*Do that millions and millions of times.*0682

*That is what we had to do here.*0685

*We also have to do that here, we want the entire population of y pulled out samples and plotted the means until we got this distribution of means.*0687

*Imagine pulling out a mean from here randomly and then finding the difference of those means and plotting that difference down here.*0699

*Do that over and over again.*0715

*You would start to get a distribution of the difference of these two means. *0718

*You would get a distribution of a whole bunch of x bar - y bar.*0727

*That is what this distribution looks like and that distribution looks normal. *0734

*This is actually one of the principle of probability distributions that we have covered before.*0742

*I think we have covered it in binomial distributions.*0747

*I know this is not a binomial distribution but the same principles apply here where if you draw from two normally distributed population*0749

*and subtract those from each other you will get a normal distribution down here.*0764

*We have this thing and what we now want to find is not just the mu sub x bar or mu sub y bar, that is not what we want to find.*0769

*What we want to find is something like the mu of x bar - y bar because this is our x bar - y bar and we want to find the mu of that.*0783

*Not only that but we also want to find the standard error of this thing.*0796

*I think we can figure out what that y might be.*0800

*At least the notation for it, that would be the standard error.*0807

*Standard error always have these x bar and y bar things.*0812

*This is how you notate the standard deviation of x bar - y bar and that is called *0817

*the standard error of the difference and that is a shortcut way of saying x bar - y bar. *0829

*We could just say of the difference.*0837

*You can think of this as the sampling distribution of a whole bunch of differences of means. *0839

*In order to find this, again it draws back on probability principles but actually let us go to variance first.*0845

*If we talk about the variance of this distribution that is going to be the variance of x bar + the variance of y bar.*0856

*If you go back to your probability principles you will see why.*0869

*This from this we could actually figure out standard error by square rooting both sides.*0874

*We are just building on all the things we have learned so far. *0881

*We know population. *0888

*We know how to do the SDOM.*0889

*We are going to use two SDOM in order to create a sampling distribution of differences.*0891

*Let us talk about the rules of the SDOD and these are going to be very, very similar to the CLT.*0898

*The first thing is this, if SDOM for x and SDOM for y are both normal then the SDOD is going to be normal too.*0909

*Think about when these are normal?*0919

*These are normal if your population is normal.*0922

*That is one case where it is normal.*0924

*This is also normal when n is large.*0927

*In certain cases, you can assume that the SDOM is normal, and if both of these have met those conditions, *0929

*then you can assume that the SDOD is normal too.*0939

*We have conditions where we can assume it is normal and they are not crazy. *0942

*There are things we have learned.*0949

*What about the mean?*0951

*It is always shape, center, spread.*0953

*What about the mean for the SDOD?*0956

*That is going to be characterized by mu sub x bar - y bar.*0959

*That is the idea.*0972

*Let us consider the null hypothesis and in the null hypothesis usually the idea is they are not different like nothing stands out.*0975

*Y does not stand out from x and x does not stand out from y.*0987

*That means we are saying very similar.*0991

*If that is the case we are saying is that when we take x bar – y bar and do it over and over again, on average, the difference should be 0.*0994

*Sometimes the difference will be positive. *1009

*Sometimes the difference will be negative.*1012

*But if x and y are roughly the same then we should actually get a difference of 0 on average.*1014

*For the null hypothesis that is 0.*1022

*The so what would be the alternative hypothesis?*1027

*Something like the mean of the SDOD is not 0. *1031

*This is in the case where x and y assume to be same.*1037

*That is always with the null hypothesis.*1051

*They assume to be the same. *1055

*They are not significantly different from each other.*1056

*That is the mean of the SDOD.*1058

*What about standard error?*1062

*In order to calculate standard error, you have to know whether these are independent samples or not.*1064

*Remember to go back to sampling, independent samples is where you know that these two *1073

*come from different populations and the picking one does not change the probabilities of picking the other.*1079

*As long as these are independent samples, then you can use these ideas of the standard error. *1089

*As we said before, it is easier when I think about the variance of the SDOD first because that rule is quite easy.*1096

*The variance of SDOD, so the variance is going to be just the variance of the SDOM + the variance of the SDOM for the other guy.*1105

*And notice that these are the x bars and the y bars.*1121

*These are for the SDOM they are not for the populations nor the samples.*1131

*From here what you can do is sort of justice derive the standard error formula.*1137

*We can just square root both sides.*1149

*If you wanted to just get standard error, then it would just be the square root of adding each of these variances together.*1153

*Let us say you double-click on this guy, what is inside of him?*1168

*He is like a stand in for just the more detailed idea of s sub x / n sub x.*1175

*Remember when we talk about standard error we are talking about standard error = s / √n.*1193

*The variance of the SDOM =s*^{2} /n.1205

*If you imagine squaring this you would get s/n but we need the variance.*1210

*We need to add the variances together before you square root them.*1220

*Here we have the variance of y / n sub y.*1224

*You could write it either like this or like this.*1235

*They mean the same thing. *1240

*They are perfectly equivalent.*1242

*You do have to remember that when you have this all under the square root sign, *1244

*the square root sign acts like a parentheses so you have to do all of this before you square root.*1253

*That is standard error.*1261

*I know it looks a little complicated, but they are just all the principles we learned before, *1265

*but now we have to remember does it come from x or does come from y distributions.*1273

*That is one of the few things you have to ask yourself whenever we deal with two samples.*1279

*Now that we know the revised CLT for this sampling distribution of the differences, *1287

*now we need to ask when can we construct a confidence interval for the difference between two means?*1298

*Actually these conditions are very similar to the conditions that must be met when we construct an SDOM.*1306

*There are a couple of differences because we are dealing with two samples.*1314

*The three conditions have to be met.*1318

*All three of these have to be checked.*1321

*One is independence, the notion of independence. *1323

*The first is this, the two samples we are randomly and independently selected from two different populations.*1329

*That is the first thing you have to meet before you can construct this confidence interval.*1340

*The second thing is this, this is the assumption for normality.*1348

*How do we know that the SDOD is normal. *1355

*It needs to be reasonable to assume that both populations that the sample comes from the population are normal or your sample size is sufficiently large.*1358

*These are the same ones that apply to the CLT.*1372

*This is the case where we can assume normality for the SDOM but also the SDOD.*1376

*In number 3, in the case of sample surveys the population size should be at least 10 times larger than the sample size for each sample.*1384

*The only reason for this is we talked before about replacement, a sampling with replacement versus sampling not with replacement.*1397

*Well, whenever you are doing a sample you are technically not having replacement *1409

*but if your population is large enough then this condition actually makes it so that you could assume that it works pretty much like with replacement.*1413

*If you have many people then it does not matter.*1427

*That is the replacement rule.*1430

*Finally, we could get to actually finding the confidence interval.*1433

*Here is the deal, with confidence interval let us just review how we used to do it for one mean.*1444

*One mean confidence interval.*1450

*Back in the day when we did one mean and life was nice and what we would do is often take the SDOM *1455

*and assume that the x bar, the sample mean is at the center of it and then we construct something like 95% confidence interval.*1466

*These are .025 because if this is 95% and symmetrical there is 5% leftover but it needs to be divided on both sides.*1484

*What we did was we found these boundary values by using this idea, this middle + or – how many standard errors you are away.*1496

*We used either t or z.*1525

*I’m just going to use t from now on because usually we are not given the standard deviation of the population × the standard error.*1529

*That was the basic idea from before and that would give us this value, as well as this value.*1530

*We could say we have 95% confidence that the population mean falls in between these boundaries.*1537

*That is for one mean.*1545

*What about two means?*1548

*In this case, we are not going to be calculating using the SDOM anymore.*1549

*We are going to use the SDOD.*1560

*If this mean is going to be x bar, this sample mean then you can probably assume that *1562

*it might be something as simple as a difference between the two means.*1575

*That is what we assume to be the center of the SDOD.*1580

*Just like before, whatever level of confidence you need.*1583

*If it is 99% you have 1% left over on the side.*1593

*You have to divide that 1% in half so .5% for the side and .5% for that side.*1598

*In this case, let us just keep the 95%.*1603

*What we need to do is find these borders.*1611

*What we can to just use the exact same idea again.*1618

*We could use that exact same idea because we can find the standard error of this distribution.*1624

*We know what that is.*1629

*Let me write this out.*1631

*We will write s sub x bar.*1640

*We can actually just translate these ideas into something like this. *1645

*That would be taking this, adding or subtracting how many jumps away you are, like the distance you are away.*1652

*That would be something like x bar - y bar but instead of just having x in the middle we have this thing in the middle.*1661

*+ or – the t remains the same, t distributions but we have to talk about how to find degrees of freedom for this guy.*1670

*The new SE, but now this is the SE of the difference.*1680

*How do we write that?*1691

*X bar - y bar + or - the t × s sub x bar = y bar.*1694

*If we wanted to we could take all that out into the square root of variance of the SDOM for x and variance of SDOM for y.*1707

*We could unpack all of this if we need to but this is the basic idea of the confidence interval of two means.*1719

*In order to do this I want you to notice something.*1727

*Here we need to find t and because we need to find t we need to find degrees of freedom *1732

*but not just any all degrees of freedom because right now we have 2 degrees of freedom. *1740

*Degrees of freedom for x and degrees of freedom for y.*1744

*We need a degrees of freedom for the difference.*1747

*That is what we need.*1751

*Let us figure out how to do that.*1753

*We need to find degrees of freedom.*1756

*We know how to find degrees of freedom for x, that is straightforward. *1760

*That is n sub x -1 and degrees of freedom for y is just going to be n sub y -1.*1764

*Life is good.*1771

*Life is easy.*1772

*How do we find the degrees of freedom for the difference between x and y?*1773

*That is actually going to just be the degrees of freedom for x + degrees of freedom for y.*1778

*We just add them together.*1790

*If we want to unpack this, if you think about double-clicking on this and get that.*1792

*N sub x - 1 + n sub y -1.*1797

*I am just putting that parentheses as you could see the natural groupings but obviously you could *1804

*do them in any order because you could just do them straight across this adding and subtracting. *1810

*They all have the same order of operation.*1816

*That is degrees of freedom and once you have that then you can easily find the t.*1820

*Look it up in the back of your book or you can do it in Excel.*1830

*Let us interpret confidence interval. *1833

*We have the confidence interval let us think about how to say what we have found.*1837

*I am just going to briefly draw that picture again because this picture anchors my thinking.*1844

*Here is our difference of means.*1852

*When you look at this t, think of this as the difference of two means.*1858

*I guess I could write DOTM but that would just be DOM.*1863

*Here what we found, if we find something like a 95% confidence interval that means we have found these boundaries.*1869

*We say something like this. *1887

*The actual difference of the two means of the real population, of the population x and y.*1891

*The real population that they come from should be within this interval 95% of the time or something like *1919

*we have 95% confidence that the actual difference between means of the population of x and population of y should be within this interval.*1939

*That comes from that notion that this is created from the SDOM.*1950

*Remember the SDOM, the CLT says that their means or the means of the population.*1955

*We are getting the population means drop down to the SDOM and from the SDOM we get this.*1962

*Because of that we could actually make a conclusion that goes back to the population.*1970

*Let us think about if 0 is not in between here.*1980

*Remember the null hypothesis when we think about two means is going to be something like this.*1987

*That the mu sub x bar – y bar is going to be equal to 0. *1993

*This is going to mean that on average when you subtract these two things the average is going to be 0.*1998

*There is going to be no difference on average.*2004

*The alternative hypothesis should then be the mean of these differences should not be 0.*2006

*They are different.*2015

*If 0 is not within this confidence interval then we have very little reason to suspect that this would be true.*2016

*It is a very little reason to think that this null hypothesis is true.*2026

*We could also say that if we do not find 0 in our confidence interval that we might in my hypothesis testing be able to also reject the null hypothesis.*2030

*But we will get to that later.*2040

*I just wanted to show you this because the confidence interval here is very tightly linked to the hypothesis testing part.*2042

*They are like two side of the same coin.*2050

*That universe is fairly straightforward but I feel like I need to cover one other thing because sometimes this is emphasized in some books.*2052

*Some teachers emphasize this over other teachers and so I'm going to talk to you about SPOOL because this will come up.*2065

*One of the things I hope you noticed was that in order to find our estimate of SDOM, *2076

*in order to find the SDOD sample error what we did was we took the variance of one SDOM *2085

*and added that to the variance of the other SDOM and square root the whole thing.*2106

*Let me just write that here. *2110

*The s sub x bar - y bar is the square root of one the variances + the variance of the other SDOM.*2111

*Here what we did was let us just treat them separately and then combine them together.*2129

*That is what we did.*2137

*Although this is an okay way of doing it, in doing this we are assuming that they might have different standard deviations.*2138

*The two different populations might have two different standard deviations.*2154

*Normally, that is a reasonable assumption to make.*2159

*Very few populations have the exact standard deviation.*2162

*For the vast majority of time because we just assumed if you come from two different population you probably have two different standard deviations.*2166

*This is pretty reasonable to do like 98% of the time.*2177

*The vast majority of time.*2182

*But it is actually is not as good as the estimate of this value then, if you had just used up a POOL version of the standard deviation.*2184

*Here is what I mean.*2198

*Now we are saying, we are going to create the standard deviation of x.*2198

*You are going to be what we used to create the standard deviation of y.*2206

*Just of not make that explicit.*2210

*I am going to write this out so that you could actually see the variance of x and the variance of y.*2213

*We use x to create this guy and we use y to create that guy and they remain separate. *2228

*This is going to take a little reasoning.*2235

*Think back if you have more data then your estimate of the population standard deviation is better, more data more accurate. *2239

*Would not it be nice if we took all the guys from the x pool and all the guys from the y pull and put them together.*2253

*Together let us estimate the standard deviation.*2262

*Would not that be nice?*2267

*Then we will have more data and more data should give us a more accurate estimate of the population.*2268

*You can do that but only in the case that you have reason to think that the population of x has a similar standard deviation to the population of y.*2278

*If you have a reason to think they are both normally distributed.*2293

*Let us say something like this.*2299

*If you have reason to believe that the population x and y have similar standard deviation *2303

*then you can pull samples together to estimate standard deviation.*2324

*You can pull them together and that is going to be called spull.*2347

*There are very few populations that you can do this for.*2351

*One thing something like height of males and females, height tends to be normally distributed and we know that.*2357

*Height of Asians and Latinos or something, but there are a lot of examples that come to mind where you could do this.*2365

*That is why some teachers do not emphasize it but I know that some others do so. *2374

*That is why I want to definitely go over it. *2378

*How do you get spull and where does it come in?*2380

*Here is the thing, in order to find Spull, what we would do is we would substitute in spull for s sub x and s sub y.*2384

*Instead of two separate estimates of standard deviations use Spull.*2396

*We will be using Spull*^{2}.2408

*How do we find Spull*^{2}?2411

*In order to find Spull*^{2}, what you would do is you would add up all of the sum of squares.2415

*The sum of squares of x and sum of squares of y, add them together and then divide by the sum of all the degrees of freedom.*2432

*If I double-click on this, this would mean the sum of squares of x + the sum of squares of y ÷ degrees of freedom x + degrees of freedom y.*2442

*This is what you need only to do in order to find Spull and then what you would do is substitute in s(x)*^{2} and s sub y^{2}.2457

*That is the deal.*2469

*In the examples that are going to follow, I am not going to use Spull because there is very little reason usually to assume that we can use Spull.*2471

*And but a lot of times you might hear this phrase assumption of homogeneity of variance.*2483

*If you could assume that these guys have a similar variance, if you can assume *2490

*they have similar homogeneous variance then you can use Spull.*2502

*For the most part, for the vast majority of time you cannot assume homogenous variance.*2508

*Because of that we will often use this one. *2514

*However, I should say that some teachers do want you to be able to calculate both.*2517

*That is the only thing.*2525

*Finally I should just say one thing. *2528

*Usually this works just as well as pull.*2531

*It is just that there are sometimes we get more of a benefit from using this one.*2536

*If worse comes to worse, and after the statistics class you are only remember this one.*2543

*If not all you are pretty good to go.*2548

*Let us go on to some examples.*2551

*A random sample of American college students was collected to examine quantitative literacy.*2556

*How good they are in reasoning about quantitative ideas.*2562

*The survey sampled 1,000 students from four-year institutions, this was the mean and standard deviation.*2565

*800 from two-year institutions, here is the mean and standard deviations.*2571

*Are the conditions for confidence intervals met?*2576

*Also construct a 95% confidence interval and interpret it.*2581

*Let us think about the confidence interval requirements.*2586

*First is independent random samples.*2593

*It does say random sample right and these are independent populations.*2596

*One is for your institutions, one is to your institutions. *2603

*There are very few people going to both of them at the same time.*2606

*First one, check.*2609

*Second one, can we assume normality either because of the large n or because we know that both these populations are originally normally distributed?*2612

*Well, they have pretty large n, so I am going to say number 2 check.*2622

*Number 3, is this sample roughly sampling with replacement?*2627

*And although 1000 students seem a lot, there are a lot of college students.*2635

*I am pretty sure that this meets that qualification as well.*2640

*Go ahead and construct the 95% confidence interval.*2643

*Well, it helped to start off with the drawing of SDOD just to anchor my thinking.*2648

*And this mu sub x bar - y bar we could assume that this is x bar - y bar.*2656

*That is what we do with confidence intervals. *2667

*We use what we have from the samples to figure out what the population might be.*2670

*We want to construct a 95% confidence interval.*2678

*That is going to be .025 and then maybe it will help us to figure out the degrees of freedom so that we will know the t value to use.*2685

*Let us figure out degrees of freedom.*2703

*It is going to be the degrees of freedom for x and I will call x the four-year university guys and the degrees of freedom for y the two-year university guys.*2706

*That is going to be 999 + 799 and so it is going to be 1800 - 2 = 1798.*2718

*We have quite large degrees of freedom and let us find the t for this place.*2747

*We need to find is this and this.*2755

*Let us find the t first. *2760

*This is the raw score, this is the t, and let me delete some of the stuff.*2765

*I will just put x bar - y bar in there and we can find that later.*2772

*The t is going to be the boundaries for this guy and the boundaries for this guy.*2782

*What is our t value?*2788

*You can look it up in the back of your book or you could do it in Excel.*2790

*Here we want to put in the t in because we have the probability and remember this one *2799

*wants two tailed probability .05 and the degrees of freedom which is 1798 = 1.896.*2806

*We will put 1.961 just to distinguish it.*2819

*Let us write down our confidence interval formula and see what we can do.*2831

*Confidence interval is going to be x bar - y bar.*2838

*The middle of this guy + or - t × standard error of this guy.*2844

*That is going to be s sub x bar - y bar.*2854

*It would be probably helpful to find this thing.*2858

*X bar - y bar.*2862

*X bar - y bar that is going to be 330 – 310.*2868

*Let us also try to figure out the standard error of SDOD which is s sub x bar - y bar.*2883

*What I'm trying to do is find this guy.*2911

*In order to find that guy let us think about the formula. *2918

*I'm just writing this for myself. *2921

*The square root of the variance of x bar + the variance of y bar .*2925

*We do not have the variance of x bar and y bar.*2937

*Let us think about how to find the variance of x bar.*2943

*The variance of x bar is going to be s sub s*^{2} ÷ n sub x.2947

*The variance of y bar is going to be the variance of y*^{2} ÷ n sub y.2959

*I wanted to write all these things out just because I need to get to a place where finally I can put in s.*2977

*Finally, I can do that.*2986

*This is s sub x and this is s sub y.*2988

*I can put in 111*^{2} ÷ n sub x which is 1000 and I could put in the standard deviation of y^{2} ÷ 800.2990

*I have these two things and what I need to do is go back up here and add these and square root them.*3017

*Square root this + this.*3028

*I know that this equal that.*3034

*We have our standard error, which is 4.49 and this is 20 + or - 1.961. *3038

*Now I could do this.*3064

*I will going to take that in my calculator as well.*3066

*The confidence interval for the high boundary is going to be 20 + 1.961 × 4.49 *3069

*and the confidence interval for the low boundary is going to be that same thing.*3085

*I am just going to change that into subtraction.*3097

*11.20.*3101

*Let me move this over.*3105

*It is going to be 28.8.*3110

*Let me get the low end first.*3117

*The confidence interval is from about 11.2 through 28.8.*3121

*We have to interpret it.*3127

*This is the hardest part for a lot of people.*3130

*We have to say something like this.*3133

*The true difference between the population means 95% of the time is going to fall in between these two numbers.*3136

*Or we have 95% confidence that the true difference between the two population means fall in between these two numbers.*3146

*Let us go to example 2.*3154

*This will be our last example.*3157

*If the sample size of both samples are the same, what would be the simplified formula for standard error of the difference?*3159

*If in addition, the standard deviation of both samples are the same, what would be the simplified formula for standard error of the difference?*3167

*This is just asking depending on how similar the two examples are can we simplify a formula for standard error.*3175

*We can.*3183

*Let us write the actual formula out so that would just x bar – y bar = square root of the variance of x bar + variance of y bar.*3184

*If we double-click on these guys that would give the variance of x / n sub x + the variance of y / n sub y.*3207

*It is asking, what if the sample size for both samples are the same?*3223

*What would be the simplified formula?*3230

*That is saying that if n sub x = n sub y then what would be this?*3231

*We can get the variance of x + variance of y / n.*3240

*Because the n for each of them should be the same.*3251

*This would make it a lot simpler.*3254

*If in addition a standard deviation of both samples are the same right then this would mean that *3260

*because the standard deviation is the same then the variances are the same.*3272

*That would be that case.*3276

*If in addition this was the case, then you would just get 2 × s*^{2} whatever the equal variances /n.3279

*That would make it a simple formula.*3294

*That would make life a lot easier but that is not always the case.*3298

*If it is you know that it will be simple for you. *3303

*That is it for the confidence intervals for the difference between two means.*3307

*Thank you for using www.educator.com.*3312

*Hi and welcome to www.educator.com.*0000

*We are going to be talking about hypothesis testing for the difference between two independent means.*0001

*We are going to go over the goal of hypothesis testing in general.*0005

*We have only looked at it for one means so far, but we are going to look at *0012

*how it changes just very suddenly when we talk about two means.*0015

*We are going to re-talk about the sampling distribution of the difference between two means.*0019

*You have just watched the confidence interval for two means, then you do not need to watch this one.*0025

*You do not need to watch that section.*0032

*We are going to talk about the same conditions for doing hypothesis testing as first confidence interval.*0034

*They need to meet three conditions before you could do either of these two.*0043

*When we talk about the modified steps of hypothesis testing for two means and the formulas that go with those steps.*0047

*Let us talk about the goal of hypothesis testing.*0055

*In one sample what we wanted to do was reject the null if *0060

*we got a sample that was significantly different from the hypothesized mu.*0065

*For instance, significantly lower or significantly higher.*0073

*A significant does not mean important like it does in our modern use of the word. *0076

*It actually means does it standout?*0083

*Is it weird enough?*0086

*Does it stand out from the hypothesized mu?*0088

*In those cases we reject the null.*0091

*Our goal is to reject the null. *0095

*We can only say whether something is sufficiently weird w cannot say whether it is sufficiently similar.*0097

*Experiment is actually a success if they reject the null.*0106

*If they do not reject the null it is considered a null experiment or what we think of as uninformative which is not actually true. *0110

*That is how traditionally is that.*0118

*This is the case where we only have one sample and we have a hypothesized population. *0123

*Here we have two samples and in order to reject the null we need to get samples that are significantly different from each other.*0130

*They stand out from each other so x is different from y, y is different from x.*0144

*That is what we are really looking for.*0151

*Once again, just like the one sample, we cannot say whether they are sufficiently similar, *0154

*but we can say whether they are sufficiently different.*0159

*It is okay if x is significantly lower than y or significantly higher.*0163

*We do not really care.*0170

*We just care about significantly different.*0171

*If you do not care about which direction these are called two-tailed hypotheses. *0173

*Let us think if x and y are different from each other then x - y should not be 0. *0179

*But if x and y are exactly the same, x = y then x – y =0.*0189

*Because you can think about this as x – x because x – y.*0196

*If you want to think about it algebraically even if you add y to each side you would get perfectly x= y.*0201

*If x and y were the same, we should expect their difference to be 0.*0211

*Let us just review very briefly the sampling distribution of the difference between two means.*0218

*This is the case where we do not know what the population is like, *0228

*but because of the CLT we actually end up knowing quite a bit about the SDOM.*0233

*This is x the population of x and population of y.*0242

*This is the SDOM of x bar, so the whole bunch of x bars and this is the SDOM for y which is a whole bunch of y bars.*0247

*We know some things about these guys and we also know we can figure out the standard error from the sample.*0258

*What is nice about this if we do not need to know anything about the population. *0280

*All we have to do is know the standard deviation of the sample which we could easily calculate *0284

*in order to estimate the standard error of these two populations. *0288

*Once we have that now we can start talking about the SDOD (the sampling distribution of the difference between means).*0294

*What we want to do is instead of finding mu sub x or mu sub y, we want to know mu sub x bar – y bar.*0306

*Here you have to think of pulling out one sample from here and one sample from here getting the difference and plotting it.*0322

*If these guys are normal, we can assume this one to be normal.*0332

*Not only that but we can figure out the standard error of this guy as well just *0336

*from knowing these because the standard error is going to be square roots of s sub x*^{2}.0342

*The variance of s/n sub x + variance of y/ n sub y.*0357

*These are all things that we have.*0366

*We do not need anything special.*0368

*We do not need sigma or anything like that.*0370

*We just need samples in order to calculate this.*0372

*If these two distributions or if these two distributions, the population distribution, *0374

*if we have a reason to suspect that these have homogeneous variance.*0384

*If their variances are the same then instead of s sub s*^{2} and s sub y^{2}, 0389

*we can actually use spull*^{2} but we would not be doing that in this lesson, but you can.0395

*Remember the rules of the SDOD are very similar to the CLT and if the SDOM for x is normal *0405

*and SDOM for y is normal then SDOD is normal too.*0415

*There is two ways that this could be true. *0419

*The first way is if populations are normal.*0421

*If population of x and y are normal then we could assume SDOM for x and y are normal.*0428

*Or are your other possibility is if n is large enough.*0435

*We want to talk about the mean for the null hypothesis.*0443

*The null hypotheses is saying that the population of x and population of y, *0450

*the difference between them is going to be 0 because they are similar.*0457

*The null hypotheses is saying both are similar, which means that the means of *0461

*the sampling distribution of the means, the SDOM means is going to be similar.*0467

*Which means that is strap in and will give us 0.*0474

*The null hypothesis says the mean of these differences of means it is going to be 0.*0478

*That is the null hypotheses and that is really saying that the SDOM for x and SDOM for the y are very similar.*0486

*Let us talk about standard error for independent samples.*0497

*Remember, we are still talking just about independent samples.*0502

*When variance is homogenous that is only used as Spull idea.*0506

*That means that x sub x bar - y bar is going to be equal to and pretend you are *0511

*writing just the regular idea where you are dividing by n sub x and n sub y.*0521

*Instead of using the variance from x and the variance from y, we are going to use that pulled variance idea.*0529

*That is going to be s pulled. *0536

*Some people think why do we just put that on top and put n sub x and n sub y at the bottom?*0547

*That will be algebraically wrong because remember, these are the denominators we would have *0554

*to have common denominators in order for us to put these together and we do not have common denominators yet.*0559

*What about in the case where variance is not homogenous and this is the vast majority of time and when in doubt, *0565

*when you do not know anything about the variance of the population go with this one. *0576

*It is just a safer option. *0582

*This is going to mean that this standard error is represented by the variance of x /n + variance of y /n.*0584

*Add these together and square the whole thing.*0602

*Just to recap, same conditions must be met in order to do hypothesis testing *0605

*for two means as the conditions for doing a confidence interval for two means.*0616

*It is that the two samples were randomly and independently selected from two different populations, *0622

*it is reasonable to assume that both populations that the sample come from are *0632

*normally distributed or the sample sizes are sufficiently large. *0636

*This was to ensure the normality of the SDOM.*0641

*Also in the case of the sample surveys, the population size should be at least 10 times larger than the sample size for each sample.*0643

*That is just assume so that we could assume replacement because probability actions change when you do not assume replacement.*0651

*Let us go in the steps of the hypothesis testing.*0663

*These are the same steps as you did when you have one mean, except now that we are subtly changing a few things.*0669

*I'm going to highlight those changes as we go through this.*0677

*First we need to state our hypotheses and remember now instead of having just the hypotheses that *0679

*the mean of the population equals this, what we are saying is that the mean of x,*0686

*population of x and the mean of the population of y those are the same. *0696

*Mu sub x - y will be 0.*0701

*You can also write it as mu sub x = mu sub y.*0707

*The alternative is that they are different from each other in some way.*0712

*Then we pick a significance level. *0718

*How different do these two populations have to be for us to say they are different?*0721

*We set a decision stage, but instead of drawing the SDOM now we draw the SDOD.*0726

*Because now we are looking at the differences between these to means. *0734

*We identify critical limits and rejection regions. *0739

*We also find the critical test statistic, the boundaries.*0743

*In order to do this we have to find the degrees of freedom for the difference.*0747

*We cannot just use the degrees of freedom for 1, degrees of freedom for the other but we actually add them together.*0753

*And then use the samples and the SDOD to compute the mean difference.*0759

*We are not just computing mean, but we are computing mean difference test statistics, as well as the p value.*0764

*And then we compare the sample to the hypothesized population.*0773

*We either reject the null or not.*0779

*We reject the null if our test statistic and p value lie in those zones of rejection.*0781

*It is like these are the weirdo zone.*0792

*This is all we know that our sample is really different from this population. *0794

*Let us talk about the different formulas that go along with these steps.*0799

*Remember the first step is going to be, what is the hypothesis, the null hypotheses, as well as the alternative.*0806

*This is not really a formula, but it is helpful to remember that this is what we really mean versus x bar – y bar does not equal 0. *0817

*This is often what is going to be the case and you can rewrite this as mu sub x bar – mu sub y bar sometimes, *0836

*but there are some mathematical ideas that you have to learn before you can write that.*0846

*I will leave that aside for now. *0857

*Second thing is significance level.*0859

*Here there are no formulas but you should know that when we say alpha= .05 we are talking about that false alarm rate.*0862

*This is the rate of rejecting the null when the null is actually true.*0873

*This is a very low rate of false alarms.*0877

*When we say alpha = .05 it is not that we calculated it but it is just that *0881

*by convention science tends to say this is the reasonable level of significance.*0887

*Sometimes people are more conservative than 1.0 or 1.001.*0895

*Number 3, we need to set that decision stage.*0900

*It is helpful to draw the SDOD and it is helpful to have our hypothesized population here. *0905

*Mu sub x bay – y bar = 0.*0924

*We assume that this point is 0.*0930

*One thing you probably also want to know about the SDOD is the formula for standard error. *0932

*The formula for standard error of the SDOD we written this a lot of times, *0941

*is the variance of x / n sub x + the variance of y / n sub y.*0951

*Another thing, you probably want to know is that we need to find these critical t.*0959

*We need to find the t values here and in order to find that you will need to know *0965

*the degrees of freedom for the difference and it is pretty easy. *0973

*It is the degrees of freedom for x + the degrees of freedom for y.*0979

*To find this, it is n sub x -1.*0983

*To find that it is n sub y -1.*0988

*We could write this as n sub x -1 + n sub y -1.*0990

*You could write it like that and then I think that is all you need to know for the decision stage.*1002

*Step 4, if you have to compute the samples mean difference you need to calculate its test statistic as well as its p value. *1011

*Remember we are going to be using t from here on out because obviously we are using s instead of sigma.*1039

*Let us talk about how to come to the sample t.*1046

*Let me write this as sample t.*1050

*The sample t is really the distance between where our sample differences versus the hypothesized difference.*1058

*We do not want it just in terms of that raw distance, we want in terms of the standard error.*1069

*It is going to be whatever our x bar - y bar is the actual sample difference -0.*1075

*That is our hypothesized population divided by the standard error s sub x bar – y bar.*1085

*That will give you how many standard errors away our actual mean difference is from 0.*1097

*Once you have this t value and you have the degrees of freedom, *1104

*then you can find the p value and then you could reject or accept the null hypotheses.*1113

*Reject or do not reject, that is really the technical idea there.*1121

*Let us go onto some examples.*1126

*The Cheesy Cheesy cookies company wanted to know whether they should have a coarse or fine texture in their cheesy cookies.*1131

*They assembled a series of taste testing panels that tasted either the coarse *1140

*or fine textured cookies and gave it a palatability score.*1143

*The higher score the better.*1153

*Is there a statistical difference in the mean palatability score between the two texture levels?*1154

*If you download the examples below and you look under the example 1, you should see a data set that looks like this.*1162

*This is the palatability score and this is the texture.*1174

*I believe that 0 = coarse and 1= fine, just so that we can make some sort of recommendation at the end.*1177

*Here we go, we have these different sets of scores, so this is the score that *1200

*one panel came up with and that panel tasted coarse textured cheesy cookies. *1209

*This panel also tasted coarse and that is the score it gave it.*1214

*Let us go up to fine.*1221

*They tasted fine texture and they give it that score. *1223

*They also tasted fine and they give it that score.*1227

*You could go and see what the different scores are and what texture they had.*1231

*First, let us think about what our x and y?*1240

*What are our two independent samples?*1245

*The two independent samples here seem to come from the two different textures.*1247

*One group of scores they all tasted coarse texture cheesy cookies.*1251

*The other group of scores tasted fine textured cheesy cookies.*1260

*It might be helpful to us to sort this data by texture.*1264

*I am going to take this and I am going to ask.*1270

*It would work if I move score over.*1281

*What I am going to do is just hit sort.*1291

*Here these are all our coarse cheesy cookie, the palatability scores and here are my fine cheesy cookie palatability scores.*1296

*Let us think about how we want to approach this problem.*1311

*First thing we want to do is create some sort of hypothesize population.*1315

*Our hypothesize population is really going to say that the coarse and *1322

*fine textured cheesy cookies there is really no difference between them. *1327

*They are the same.*1330

*The mu sub x bar - y bar should equal 0.*1332

*The alternative is that they are different from each other in some way. *1337

*We do not know which one taste better.*1346

*Let us just be neutral and say we do not know whether the coarse cheesy cookies *1352

*are better than the fine or to fine cheesy cookies are better than the coarse.*1358

*We want to know whether these palatability scores are different or the same.*1364

*Let us set a significance level for how different they have to be.*1370

*Our significance level could be alpha= .05.*1377

*Finally let us set a decision stage.*1386

*Here I am going to draw SDOD, can we assume normality?*1390

*Well, they are different and let us look here.*1398

*We have 8 scores and 8 scores, the n is low.*1405

*Technically, we might not be able to do hypothesis testing.*1416

*Let us say for some reason that your teacher wants you doing anyway. *1424

*But one of the things that should come up when you see low n like this is that you should question *1430

*whether hypothesis testing is the right way to go because it may not reflect the conditions *1436

*that we need to have set before we can assume all the stuff.*1446

*Just for the problem solving and practice here, let us go with that.*1449

*But if you want it to be smaller you can tell your instructor the conditions are meet for hypothesis testing.*1454

*Here we set our little lower n rejection and why do we just go ahead and put in our mu here.*1466

*It is going to be 0 and it will be helpful to find out that t values out here.*1478

*Let us go ahead and do that. *1483

*What are our critical t?*1486

*Critical t or the boundaries.*1491

*In order to find the critical t, we are going to have to find the degrees of freedom, DF of differences. *1494

*N sub x we will call x coarses.*1503

*X will be coarse cheesy cookies and y will be fine.*1512

*You can use c and f if you want to.*1521

*This is going to be 8 and this is also 8.*1524

*The degrees of freedom for each of these is 7 so this is going to be 14.*1528

*That is a pretty low degrees of freedom.*1534

*That is all we can assume normality here.*1537

*Let us find the critical t.*1540

*In order to find that we would use t inverse because we have the two tailed probability .05 and we have the degrees of freedom.*1545

*This gives us a positive version.*1562

*The negative version would just be the negative of that number because they are perfectly symmetrical. *1565

*2.14 the critical t is + or -2.14.*1573

*Now that we have that, then we could go ahead and look at the actual samples themselves. *1581

*Step 4, is we need to find the samples mean difference.*1589

*We need to find x bar – y bar, but we also need to find this mean differences t.*1598

*The t sub x bar - y bar.*1606

*We need to find that as well as the p value. *1610

*Let us go ahead and do that.*1613

*We just started from step 3 and step 4 is really the mean difference and that is just the average of these guys - the average of these guys.*1618

*That is their average difference. *1656

*This is saying that the coarse scores tend to be on average lower than *1662

*the fine scores because we do course score – fine score.*1668

*We get a negative number.*1671

*The coarse score number must have been small.*1672

*Actually before we go on, it might be helpful to find the standard error of this situation.*1677

*In order to find the standard error of the difference we need to find *1690

*the square roots of the variance of x ÷ n sub x + the variance of y ÷ n sub y.*1699

*This is going to be our standard error that we need.*1717

*In order to find that it would be helpful to find each of these pieces by themselves.*1724

*I guess we could find the whole thing, the variance of x ÷ n sub x and the variance of y ÷ n sub y.*1731

*I will put each of these on different lines like we can do all of it together.*1750

*We could just add them all up here.*1754

*Let us find that.*1757

*The variance, thankfully Excel has all these functions.*1763

*Let us check and make sure that this variance will give us n-1.*1771

*The variance of x ÷ 8 and the variance of all my fine cheesy cookie values ÷ 8.*1778

*We have these two variances and when we divide by n sub x we are getting the variance of the SDOM.*1799

*If we add those together then get the square root, then we get the standard error of the difference.*1811

*The square root of these two guys added together and that is 11.16.*1820

*Here I will just add this information so the standard error of the difference =11.16.*1830

*In order to find this t, we need to have this difference between the means -0 / the standard error of the difference. *1851

*We can easily do that now. *1866

*Here in order to find the sample t we could put the mean difference -0.*1871

*If you want to keep it technical you do not need that -0 / the standard error of the difference.*1891

*Our sample t says the difference is not at 0 it is actually way down here.*1901

*It is not significantly different. *1914

*Well, one thing we could do is just operate here and compare this number to this number. *1917

*This sub boundary here is -2.14.*1923

*-4.73 is like out here so we definitely know it is way significant.*1928

*It is way standing out from the expected mean but we can also find the p value. *1935

*Now remember in Excel one of the things it needs a positive t value. *1944

*If you have a negative t value you have to turn it into a positive one, but it is okay because it is perfectly symmetrical. *1951

*The degrees of freedom that we are talking about are going to be this *1959

*new combined degrees of freedom because we are always talking that the SDOM now.*1963

*This is the degrees of freedom for this SDOD and that is 14 and it is a two-tailed hypothesis.*1969

*Our p value is .0003.*1976

*I will not write the last up here but we can just talk about it.*1981

*The last step would be we reject or do not reject the null. *1991

*Well, we reject the null here because our t value is much lower than our significance level.*1997

*Our t value, our sample t is more extreme than our critical t.*2003

*Here what we would say is that there is a statistical difference between the two texture levels.*2010

*One that is very unlikely to be attributed to by chance, because that is what this t values.*2018

*If it was by chance it would have .03% probability.*2026

*It is pretty low.*2033

*Example 2, scientists have found certain tree resins that are deadly to termites.*2035

*To test the protective power of resin protecting the tree, a lab prepared 16 dishes with 25 termites in each.*2042

*Each dish was randomly assigned to be treated with 5 mg or 10 mg of resin.*2050

*At the end of 15 days, the number of surviving termites was counted.*2055

*Assume that termites survival tends to be normally distributed with both dosage levels.*2060

*Is there a statistical significant difference in the mean number of survival for those two doses?*2066

*Now here I think it is worth than just discussing what will be our x and y.*2072

*Our x might be the 5 mg population and our y might be the 10 mg population.*2077

*The n sub x some people might think there are 25 termites but actually there is 25 termites in each of 10 Peachtree dishes.*2087

*There are 8 Peachtree dishes that have been randomly treated with 5 mg and 8 have been treated with 10 mg.*2099

*This is 8 and 8.*2109

*When I say 8, we mean the dishes of treatment and the termites are not the subject they are the cases that we are interested in.*2113

*The termites are the test.*2124

*You can get 25 termites surviving or you could get 0 surviving.*2128

*How many termites survived?*2134

*That is our dependent variable.*2135

*Okay, let us see. *2137

*Well one thing we could do is start off with our hypotheses.*2142

*Our null hypotheses is that these two dosage levels are roughly the same.*2146

*We might say something like the mu sub x bar - y bar which is equal 0 are the same.*2153

*The alternative is that they are not the same. *2161

*Maybe that one is more powerful than the other.*2166

*We do not know which one.*2169

*We could easily set our significance level to be .05.*2173

*Let us talk about the actual set up, the decision stage.*2179

*In the decision stage, let us see what we have here.*2184

*We have set up this .05 level rejection and we could just go ahead and this is the x bar - y bar, but what would be that t?*2195

*The nice thing about this being 0 is that the t distribution as well as the x bar – y bar start off the same.*2213

*They are not going to have the same numbers out here. *2226

*Okay, so that is why we do have to put them on different lines.*2229

*They are still talking about different things.*2233

*Let us talk about the t values.*2235

*Before we do, it might be helpful to figure out the new degrees of freedom.*2240

*The degrees of freedom of differences will be 7 + 7 =14.*2247

*Here we can do hypothesis testing just jump in right away because given *2255

*the termite survival tends to be normally distributed within these two dosage rates.*2261

*If you go to example 2, you will actually see the data here.*2267

*Here we see dosage and here is the 5 mg, as well as the 10 mg.*2284

*Here are the survival counts.*2293

*How many termites survived?*2294

*Notice that there is no survival count over 25.*2296

*25 is the maximum you can have, but even the highest gives me 16.*2299

*What if the survival count cannot go below 0 because we cannot have negative termite surviving.*2304

*Here we have the survival count.*2311

*Let us see what we have here.*2317

*Can we figure out what the critical t is.*2323

*Can we figure out what the critical t is?*2329

*I think we can.*2335

*Let us see.*2336

*You can use the book but I am going to use Excel to find the critical t.*2338

*I am going to write for myself step 4.*2344

*I know the two-tailed probability that I need .05 and I know my degrees of freedom is 14.*2347

*I see that the critical t is the same as before and because we use *2362

*the same two tailed probability and the same degrees of freedom of differences.*2367

*Here we know that it is -2.14, as well as positive 2.14.*2372

*What we can do is now from here go on to looking at our actual sample.*2384

*This is actually step 3, it is a part of our decision stage. *2394

*Step 4, is now actually talking about the sample. *2406

*It will help to find the sample mean difference, so that is going to be the average of one of these x - the average y.*2410

*We want to know is this is difference going to be significantly different from 0?*2431

*We cannot just look at the raw scores because we need to figure out how many standard errors away we are.*2436

*How shall we find the standard error for the difference?*2443

*That is equal to the square root of the variance of x/ n sub x + variance of y/ n sub y.*2448

*Let us find the variance of x/ n sub x over and variance of y/ n sub y.*2458

*Let us find the variance of x/8 and the variance of y /8.*2468

*We see that the variance for y is a lot different than the variance for x.*2486

*That is helpful for us to just look at briefly right now just because this will probably give us an idea *2493

*that the variance of samples are so different we probably do not have a good reason to pull these two together.*2500

*We do not have a good reason to assume that the populations are similar.*2507

*When in doubt go with non homogenous variances. *2511

*Just assume that they are different. *2518

*Once we have that then we can find the square root of adding these two standard errors together and we get 2.5.*2520

*Once we have all of that then we can find the samples mean difference t.*2535

*And that would be the samples mean difference -0 divided by the standard error of the SDOD.*2548

*What would that be?*2572

*That would be this guy and I am going to leave that subtract 0 part divided by the standard error and we get to 2.15.*2575

*We are close but it is still more extreme than 2.14.*2586

*It does not have to be extreme and the -n could be either extreme in the negative n or extreme in the positive n.*2595

*This is extreme in the positive n.*2603

*It is just right outside our borders.*2607

*Let us find the p value. *2609

*In order to find that p value we use t distribution because we have the t value that *2611

*we want the degrees of freedom and we wanted to be a two-tailed p value.*2620

*It is going to add up this little chunk and this little chunk together and that can be .049.*2625

*We will just skip step 4, our p value =.0449 that is right just a hair underneath our alpha.05.*2635

*We would probably reject the null.*2653

*Example 3, 2 months before smoking ban in bars, a random sample of bar employees were assessed on respiratory health.*2657

*Two months after the ban, another random sample of employees were assessed.*2672

*Researchers saw a statistically significant increase in the mean scores of health.*2678

*P= .049 we had an example of that two tailed.*2684

*Which of the following is the best interpretation for this result?*2689

*The probability is only .049 that the mean score for all of our employees increased from before to after the ban.*2693

*Is that what this means?*2706

*For me it helps to draw that SDOD and it is saying the null hypotheses would be *2708

*the same like before and after are the same.*2715

*What they actually found is that there is some extreme value.*2720

*There is the increase in mean scores.*2727

*There is a positive difference from after – before.*2735

*There is the increased.*2742

*It is somewhere up here, that increase tells us that.*2745

*P= .04.*2749

*We can actually draw this carefully, it is just right above that cut off.*2753

*There is only .049 probability that the mean score for all bar employees increase.*2760

*That is not what this means.*2775

*It is not saying that there is only a small chance that it increase.*2778

*It is actually saying there is a pretty good chance that it is not the same.*2783

*There is a pretty small chance that it is the same.*2787

*This one we can just rule out.*2792

*Another possibility is that the mean score for all bar employees increased by more than 4.9%.*2796

*Does this p value actually talk about the raw score on respiratory health?*2805

*It does not talk about that score at all, it is the probability of finding such a difference.*2814

*It does not have anything to do with actual scores. *2821

*What about this one?*2825

*An observed difference in the sample means as large or larger than the sample is unlikely to occur *2828

*if the mean score for all bar employees before and after the ban were the same.*2835

*This actually have something we can use.*2839

*This is about considering that the means score for before and after are the same. *2842

*That is important because that is what the SDOM actually represents.*2851

*That is what this p value is actually talking something about this idea that when we get the sample, *2854

*we consider that they were just the same.*2865

*This is saying an observed difference in sample means as large or larger than a sample is very unlikely to occur.*2867

*It is likely to occur with .049% if the mean score for all bar employees the true score is actually the same.*2876

*This is a pretty good contender because the SDOD is talking about how .049 means very unlikely.*2889

*This I would leave as a definite contender. *2900

*Maybe there is a better answer.*2902

*There is a 4.9% chance that the mean score of all bar employees after the ban is actually lower than before the ban.*2905

*There is a small chance of the opposite hypotheses picture that is probably not the case.*2915

*It depends on what the null hypothesis was.*2925

*The null hypothesis and a two mean hypotheses test is usually the same not the one is less than the other.*2934

*We do not usually do that.*2953

*Maybe there is a way and that could be true.*2954

*It is probably not true if we did hypothesis testing at all.*2958

*Only 4.9% of the bar employees had their score drop but the other 95% had their scores increase.*2961

*This would be a correct interpretation if we are not talking about the SDOD.*2971

*If this was not a reflection of the population then maybe that would be true.*2977

*This is not talking about population, it is talking about the SDOD.*2982

*This is a wrong interpretation.*2987

*The correct answer is c.*2990

*That is our last example for hypotheses testing with two independent means.*2992

*Thank you for joining us on www.educator.com.*2998

*Hi and welcome to www.educator.com.*0000

*We are going to talk about confidence interval and hypothesis testing for the difference of two paired means.*0002

*We have been talking about independent samples so far, one example, two independent samples.*0008

*We are going to talk about paired samples.*0017

*We are going to look at the difference between independent samples and paired samples.*0020

*We are also going to try and clarify the difference between independent sample *0025

*and independent variables because paired samples still use independent variables.*0029

*We are going to talk about two types of t-tests.*0035

*One that we covered or also called hypothesis testing and one that we covered so far with independent samples.*0039

*The new one that was cover with paired samples.*0046

*We are going to introduce some notation for paired samples, go through the steps of hypothesis testing *0050

*for paired samples and adjust or add on to the rules of SDOD that we already have looked at.*0058

*Finally we are going to go over the formulas that go with the steps of hypothesis testing for independent as well as paired samples.*0069

*We are going to briefly cover confidence interval for paired samples.*0081

*Here is the goal of hypothesis testing. *0085

*Remember, with one sample our goal was to reject the null when we get a sample *0091

*that significantly different from the hypothesized population.*0098

*When we talk about two-tailed hypotheses we are really saying the *0102

*hypothesized population might be significantly higher or significantly lower. *0107

*Either way, we do not care.*0114

*The sample is too low or too high, it is too extreme in some way. *0116

*If that is the case, we reject the null.*0123

*In two samples, what we do is we reject the null when we get samples that *0125

*are significantly different from each other in some way.*0132

*Either one is significantly lower than the other or the other is significantly lower than the one.*0135

*It does not matter.*0141

*Our null hypothesis becomes this idea that x - y either = 0 because they are the same *0142

*and the alternative is that it does not equal 0 because they are different from each other.*0152

*If they are the same that is considered the null hypotheses and when they are different that considered the alternative hypotheses.*0159

*Remember another way you could write this is by adding y to each side and then you get x=y.*0168

*X = y they are the same.*0174

*In that way you know that you are covering the entire space of all the differences and the end of the day *0176

*we can figure out whether they are the same or we do not think that the they are the same.*0186

*Let us talk about independent samples versus paired samples because from here on out,*0195

*we are totally going to be dealing with paired samples. *0203

*It would help to know what those are.*0205

*Independent samples, the scores are derived separately from each other. *0208

*For instance they came from separate people, separate schools, separate dishes.*0212

*The samples are independent from each other. *0219

*My getting of the sample had nothing to do with my getting of this other sample.*0222

*In dependent, another word for paired, in dependent or paired samples the scores are linked in some way.*0227

*For instance, they are linked by the same person so my score on the math test and my score on the english test are linked because they both come from me.*0236

*Maybe we are one married couple, we ask one spouse how many children would you like to have *0248

*and you ask the other spouse how many children would you like to have?*0258

*In that way, although they come from different people these scores are linked because they come from the same married couple.*0262

*Another thing might be a pre and post tests of the class.*0269

*Maybe a statistics class might do a pre and post test.*0276

*Maybe 10 different statistics classes from all over the United States picked to do a pre and post test.*0279

*Those tests are linked because the same class did the first test and the second test.*0287

*10 different classes did the pairs.*0295

*It is not just a hodgepodge of pretests scores and a hodgepodge of posttest scores, it is more like a neat line *0298

*where the pretests scores for this guy, but for this class is lined up with the pretests scores for that class.*0309

*They are all lined up next to each other. *0317

*We know these definitions, let us see if we can pick them out. *0319

*Which of these is which?*0327

*The test scores from Professor x’s class versus test scores from professor y class.*0329

*Will these be independent samples because they just come from different classes?*0336

*They are not each score is not linked in any particular way.*0341

*River samples from 8 feet deep versus 16 feet deep.*0346

*This also does not really seem like paired samples unless they went through *0350

*some procedure to make sure it is the same spot in the river.*0355

*That is probably an independent sample.*0360

*Male heights versus female heights, they just a jumble of heights over here and a jumble of heights over here.*0364

*They are not like match to each other.*0370

*They are independent samples.*0372

*Left hand span versus right hand span will in this case basically these two spans came from the same person.*0375

*It is not a hodgepodge like left hand right hand from person 1, left hand right hand for person2 or person 3.*0384

*I would say this is a paired sample.*0392

*Productive vocabulary of two-year-old infant often raised by bilingual parents versus monolingual parents.*0395

*It is a bunch of scores here and a bunch of scores here. *0402

*They are not lined up in any way.*0406

*I would say independent.*0408

*Productive vocabulary of identical twins, twin 1, twin 2.*0410

*Here we see paired samples.*0417

*Scores on an eye gaze by autistic individual and age matched controls.*0420

*Autistic individuals often have trouble with eye gaze and in order to know that you *0427

*would have to match them with people who are the same age who are not autistic.*0432

*Here we have autistic individual lined up with somebody who is their same age is not autistic.*0438

*They are these nice even pairs and each pair has eye gaze scores.*0445

*I would say these are paired samples.*0452

*Hopefully that give you a better idea of some examples of paired samples. *0457

*What about independence samples versus independent variables?*0462

*What you will also see is IV.*0469

*In multi sample statistics like 2, 3, 4 samples we are often trying to find some *0471

*predictive relationship between the IV and the DV.*0477

*The independent variable and the dependent variable. *0481

*Usually this is often called the test or the score.*0484

*The independent variable is seen as the predictor and the dependent variable *0488

*is the thing that is been predicted the outcome.*0495

*We might be interested in the IV of parent language and you might have two levels of bilingual and monolingual.*0498

*You might be interested in how that impacts the DV of children’s vocabulary.*0519

*Here we have these two groups, bilingual and monolingual.*0534

*We have these scorers from children and these are independent samples because *0542

*although we have two groups these scores are not linked to each other in any particular way. *0550

*They are just a hodgepodge of scores here and a hodgepodge of scores here. *0556

*On the other hand, if our IV is something like age of twin.*0560

*We have slightly older like a couple of minutes or seconds, and younger.*0572

*We want to know is that has an impact on vocabulary.*0582

*We will have a bunch of scorers for older twins versus younger twins, but these scores are not just in a jumble. *0593

*They are linked to each other because these are twins.*0611

*They are identical.*0615

*This is the picture you could draw and the IV tells you how you determine these groups. The paired parts tells you whether these groups scores are linked to some scores *0617

*in the other group for some reason or another.*0640

*Here they are linked but here they are not linked.*0642

*In all t tests, we are calling them hypothesis testing.*0646

*We are going to have other hypothesis tests but so far we are using t test.*0657

*T tests always have some sort of categorical IV so that you can create different groups *0662

*and in t-tests it is always technically two groups, two means, paired means.*0668

*The DV is always continuous.*0674

*The reason that the dependent variable or the scores always continuous is because you need to calculate means in order to do a t test.*0678

*We are comparing means too and looking at standard error and you can compute mean *0687

*and standard error for categorical variables.*0694

*If you have a categorical variables such as you know, yes or no, you cannot quite compute a mean for that.*0697

*Or if you have a categorical variable like red or yellow, you cannot compute a standard error for that.*0707

*If you did have a categorical DV and a categorical IV, you would use what it is called the logistic test.*0713

*We are actually not going to cover that.*0721

*That does not usually get covered in descriptive and inferential statistics. T*0723

*Usually you have to graduate level work or higher level statistics courses.*0727

*There are two types of t test given all of this.*0735

*Remember all t tests have this.*0740

*These are all t tests.*0742

*Both of these t tests are going to use categorical IV and continuous DV.*0743

*The first kind of t test is what we have been talking about so far, independent samples t tests. *0750

*The second type is what we are going to cover today called paired or dependent samples. *0762

*Both of these have categorical IV and continuous DV.*0769

*Let us have some notations for paired samples.*0778

*Just like before, with two sample independent sample t test, for one example, *0784

*you might call it x so that its individual members are x sub 1, x sub 2, x sub 3. *0792

*Remember each sample is a set of numbers.*0797

*It is not just one number but a set of numbers.*0800

*Second sample, you might call y.*0803

*I did not have to pick x and y though.*0807

*I could pick other letters.*0809

*Y could just mean another sample.*0810

*You could have picked w or p or n.*0816

*We usually try to reserve n, t, f, d, k for other things in statistics, but it is mostly by culture more than we have to do it by rules.*0820

*Here is the third thing you need to know for paired samples.*0837

*With paired samples remember x sub 1 and y sub 1 are somehow linked to each other.*0842

*They either come from the same person or the same married couple or *0848

*they are a set of twins or it is an autistic person and age matched control.*0853

*All these reasons why these are linked to each other in some way.*0859

*And because of that you can actually subtract these scores from each other and get a set of different scores.*0865

*That is what we call d.*0872

*D is x sub 1 – y sub 1.*0874

*What is the difference between these two scores?*0877

*What is the difference between these two scores and what is the difference between these two scores?*0882

*These are paired differences.*0888

*Let us think about this.*0891

*If the mean of x is denoted as x bar and the mean of y is denoted as y bar, what do you think the mean of d might be?*0894

*I guess d bar and that is what it is.*0902

*If you got the mean of this entire set that would be d bar.*0907

*Once you have d bar, you could imagine having a sampling distribution made of d bars.*0912

*It is not x bars anymore, sampling distribution of the mean is the sampling distribution of the mean of a whole bunch of differences.*0924

*That is a new idea here.*0942

*Imagine getting a sample of d, calculating the mean d bar and placing it somewhere here.*0945

*You will get a sampling distribution of d bars.*0959

*That is what we are going to learn about next. *0964

*These are means of a bunch of linked differences.*0966

*When we go through the steps of hypothesis testing for paired samples it is going *0971

*to be very similar to hypothesis testing for independent samples with just a few tweaks. *0979

*First you need to stay to hypothesis and often our null hypothesis is that the two groups of scores, the two samples x and y are the same. *0985

*Usually that is the null hypothesis. *0997

*You put the significance level, how weird does our sample has to be for us to reject that null hypothesis.*1004

* We set a decision stage and we draw here the SDOD d bar.*1013

*We identify the critical limits and rejection regions and we find the critical test statistic.*1020

*From here on out I am going to assume that you are almost never going to *1027

*be given the actual standard deviation of the population. *1033

*From here on out I am usually going to be using t instead of z.*1038

*Then we use the actual sample differences and SDOD in order to compute the mean differences.*1041

*We are not dealing with just the means, we are dealing with mean differences, test statistics, and p value.*1053

*We compare the sample to the population and we decide whether to reject the null or not.*1061

*Things are very similar so far.*1069

*It is going to make us figure out what SDOD is all about.*1073

*The rules of SDOD we are now adding on to sampling distribution of *1083

*the differences between means that we talked about before you.*1093

*We are going to add onto that.*1100

*The SDOM for x and y are normal then the SDOD is normal too.*1103

*That is the same here.*1109

*The mean for the null hypotheses now looks like this. *1111

*Remember the SDOD with the bar, the mean here is no longer called the mu sub x bar - y bar because it is actually x bar - y bar.*1116

*A whole bunch of them and then you find the mean of them.*1132

*That is called d bar.*1136

*That is the new notation for the differences of paired samples. *1137

*Here the mu of d bar for the null hypotheses equal 0. *1147

*Remember for independent samples = that for mu sub x bar - y bar that = 0. *1153

*It is very similar.*1162

*For standard error for independent samples when various is not homogenous, which is largely the case, *1164

*what we would use is s sub x bar - y bar.*1174

*Instead here for paired samples, we would use s sub d bar.*1182

*Here what we would do is take the square root of the variance of *1188

*the standard error from x and the standard error variance of y bar and add that together.*1194

*If you wanted to write that out more fully, that would be s sub x*^{2} the variance of x / n sub x + variance of y / n sub y.1207

*That is what you would do if life was easy and you have independent samples.*1228

*That is what we know so far.*1238

*What about for paired samples?*1240

*For paired samples you have to think about the world differently. *1242

*You have to think first we are getting a whole bunch of differences then we are finding the standard error of those differences.*1245

*Here is that we are going to do.*1253

*Here we would find standard error of those differences by looking at *1256

*the standard deviation of the differences ÷ how many differences we have.*1263

*This is a little crazy, but when I show you it, it will be much more easy to understand.*1272

*I think a lot of people have trouble understanding what is difference between this and this?*1281

*I cannot keep track all these differences.*1287

*We have to draw SDOD.*1291

*You have to remember it is made up of a whole bunch of d bars.*1302

*He is made up of a whole bunch of these.*1312

*You have to imagine pulling out samples, finding the differences, *1314

*averaging those differences together, then plotting it here.*1324

*Each single sample it has a standard deviation made up of differences.*1328

*Once you plot a whole bunch of these d bars on here, this is going to have a standard deviation and that is called standard error.*1337

*Here we have mu sub d bar and this standard error is standard error sub d bar.*1347

*Standard deviations of d bar whereas this is just for one sample.*1359

*This guy is for entire sampling distribution.*1367

*Let us talk about the different formulas that go with the steps of hypothesis testing.*1378

*Hopefully we can drive home the difference between SDOD from before and SDOD now, we will call it SDOD bar.*1385

*For independent samples, first we had to write down a null hypothesis and alternative hypothesis.*1398

*Often a null hypothesis was that the mu sub x bar - y bar = 0 or mu sub x bar - y bar does not equal 0 as the alternative.*1408

*In paired samples our hypothesis looks very similar except now we are not dealing with x bar - y bars but we are dealing with difference bars. *1421

*The average of differences. *1438

*The mean differences. *1440

*This is the differences of means.*1442

*This is mean of differences.*1448

*We will get into the other one.*1453

*Mu sub d bar does not =0.*1457

*This so far it seems like okay. *1463

*Here difference of means and d bar is the mean of a whole bunch of differences.*1467

*We get a whole bunch of differences first, then we find the mean of it. *1484

*Here we find the means first and we find the difference between the means.*1489

*This part is actually the same.*1495

*It is alpha =.05 usually two tailed.*1500

*Step 2, we got that.*1510

*Significant level, we get it.*1515

*Step 3 is where we draw the SDOD here.*1517

*Here we draw the SDOD bar.*1521

*Thankfully you could draw it in similar ways, but conceptually they are talking about different things. *1530

*Here how we got it was we pulled a bunch of x. *1538

*We got the mean then we pulled a bunch of y then we got the mean and subtracted those means and plotted that here.*1543

*We did that millions and millions of time with a whole bunch of that.*1550

*We got the entire sampling distribution of differences of means.*1554

*Here what we did was we pull the sample of x and y.*1560

*We got a bunch of the differences and then we average those differences and then we plot it back.*1568

*Here this is the sampling distribution of the mean of differences.*1579

*Where the mean go in the order is really important.*1591

*Here we get mu sub x bar - y bar, but here we get mu sub d bar.*1599

*In order to find the degrees of freedom for the differences here what we did was *1607

*we found the degrees of freedom for x and add it to it the degrees of freedom for y.*1615

*We are going to do something else in order to find the degrees of freedom for *1620

*the difference we are going to count how many differences we average together and subtract 1.*1626

*This is how many n sub d – 1.*1637

*Finally we need to know the standard error of the sucker.*1644

*The standard error of differences here we called it s sub x bar - y bar and that *1650

*was the standard error of x, the variance of x bar + the variance of y bar.*1659

*The variance of these two things added together then take the square root.*1670

*This refers to this distribution with the spread of this distribution. *1676

*This difference here is actually going to be called s sub d bar and that is *1688

*going to be standard deviation of your sample of differences ÷ √n of those differences.*1696

*Last thing, I am leaving off step 5 because step 5 is explanatory.*1707

*Step 4, now we have to find the sample t.*1719

*Our sample is really two independent samples.*1723

*We have a sample of x and a sample of y.*1732

*Because of that we need to find the difference between those two means. *1734

*We find the mean of this group first, the mean of this group and we subtract.*1741

*We find the means first then we subtract - the mu sub x bar - y bar.*1747

*I want you to contrast this with this new sample t.*1756

*Here we get a bunch of x and y, we have two samples.*1761

*We find the differences first then we average.*1766

*Here we find the average first and find a different.*1773

*Here we find the differences then we find the average.*1776

*That is going to be d bar.*1782

*D bar – mu sub d bar.*1784

*This is getting a little bit cramped.*1790

*We divide all of that by the standard error of the difference and you could substitute that in.*1796

*Divide all that by the standard error of the differences.*1803

*You see how here it really matters when you take the differences.*1811

*Here you find the differences first and then you just deal with the differences.*1820

*Here, you have to keep finding means first then you find the differences between those means.*1824

*Let us talk about the confidence interval for these paired samples.*1830

*The confidence intervals are going to be very similar to the confidence intervals that you saw before with independent samples.*1841

*I am just covering it very briefly.*1849

*Let us think about independent samples.*1851

*In this case, the confidence interval was just going to be the difference of means and + or - t × the standard error.*1854

*You need to put in the appropriate standard error and use the appropriate degrees of freedom as well. *1877

*In confidence intervals for paired samples it is going to look very similar except instead of having the differences of means *1884

*you are going to put in the mean difference d bar + or - t × the standard error. *1897

*Remember standard error here is going to mean s sub x bar - y bar.*1906

*The standard error here is going to be s sub d bar.*1914

*In order to find degrees of freedom you have to take the degrees of freedom for x and add that to the degrees of freedom for y.*1918

*In order to find degrees of freedom you have to find the degrees of freedom for d *1928

*your sample of differences and that equals how many differences you have -1.*1935

*Let us talk about examples.*1945

*There is a download available for you and says this data set includes the highway *1953

*and city gas mileage for random sample of 8 cars.*1958

*Assume gas mileage is normally distributed.*1962

*It says that because we could see your sample is quite small so we do not have *1965

*a reason to assume that normal distribution of the SDOM.*1970

*Construct and interpret the confidence interval and also conduct an appropriate t test to check your confidence interval interpretation. *1974

*Here I have my example and going to example 1.*1984

*Here we have 8 models of cars, their highway miles per gallon, as well as their city miles per gallon.*1989

*You can see that there is a reason to consider these things as linked. *2004

*They are linked because they come from the same model car.*2010

*Let us construct the confidence interval.*2013

*Remember in confidence interval what we are going to do is use our sample in order to predict something about our population.*2018

*Here we will use our sample differences to say something about the real difference between these two populations.*2028

*Here is the big step of difference when you work with paired samples.*2036

*You have to first find the paired differences so the set of d.*2042

*That is going to be one of these will take highway - the city.*2048

*That x1 – y1, x2 – y2, x sub 3 – y sub 3.*2054

*Here are all our differences and we can now find the average differences.*2062

*We can find the standard deviation of these differences and all the stuff.*2067

*Let us find confidence interval and this helps me to say what I need is my d bar + or - t × the standard error.*2071

*In order to find my t but in order to do that I need to find my degrees of freedom.*2090

*My degrees of freedom is just going to be the degrees of freedom of the d.*2098

*How many differences I have -1.*2107

*That is count how many differences they should have the same number of differences as cars -1 =7. *2110

*Once I have that, I could find my t.*2121

*I also need to find d bar.*2126

*Let us find t.*2130

*I need to find t and t inverse and I probably am going to assume a 95% confidence interval.*2134

*My two tailed probability is .05 and my degrees of freedom is down here and so that will be 2.36.*2146

*Those are my outer boundaries and let us also find d bar, the average.*2157

*I almost have everything I need. *2165

*I just need standard error.*2172

*Standard error here is going to be s sub d ÷ the square root of how many differences I have.*2174

*That is going to be the standard deviation of my differences ÷ the square root of 8 because I have 8 differences.*2187

*Once I have that, then I can find the confidence interval.*2206

*The upper boundary will be the d bar + t × standard error and the lower boundary is the same thing, except that this - t × standard error.*2209

*My upper boundary is that 10.6. *2244

*My lower boundary is that 7.6.*2249

*To interpret my confidence interval I would say the real difference between highway miles per gallon *2253

*and city miles per gallon I have 95% confidence that the real difference in the population is between 10.6 and 7.6.*2264

*Notice that 0 is not included in here in this confidence interval.*2274

*It would be 0 if highway and city miles per gallon could be equal to each other by chance.*2280

*There is less than 5% chance of them being equal to each other. *2288

*Because of that, I would guess that we would also reject the null because it does not include 0.*2295

*Let us do hypothesis testing to see if we do really reject the null because it does not include 0 *2304

*I would predict that we would reject the null.*2312

*Let us go straight into hypothesis testing here.*2314

*First things first. *2317

*The step 1, the null hypothesis this should be that the mu of d bar. *2320

*Here let us do hypothesis testing. *2332

*The first step is mu sub d bar is equal to 0.*2344

*Highway and city gas mileage are the same but the alternative is that one of them is different from the other.*2356

*That they are different from each other in some way.*2366

*It is significantly stand out.*2369

*This difference stands out.*2371

*That would be that mu sub d bar does not equal 0.*2373

*Step 2, my significance level, the false alarm rate is very low .05 and two tailed.*2378

*Let us set our decision stage.*2392

*I need to draw an SDOD bar and here I put my mu as 0 because the mu sub d bar will be 0.*2397

*Let us also find the standard error here. *2418

*The standard error here is going to be s sub d bar and that is really the standard deviation of the d / √n sub d.*2421

*That I could compute here.*2434

*Actually, we already computed that because we have the standard deviation of the d bars / the square root of how many d I have.*2439

*That is .64.*2449

*What is my degrees of freedom?*2455

*That is 7 because that is how many differences I have -1.*2458

*Based on that I can find my t and my t is going to be + or - 2.36. *2466

*Let us deal with our sample. *2476

*When we talk about the sample t, what we really mean is what the x bar of our sample differences that would be d bar.*2483

*I would just put x bar sub d because it is a simpler way of doing it.*2502

*- the mu which is 0 / the standard error which is .64.*2505

*I could just put this here so I can skip directly to step 4 and I will compute my sample t.*2512

*I should say this is my critical t so that I do not get confused.*2527

*My sample t is going to be d bar - mu / standard error.*2533

*That is d bar - mu which is 0 ÷ standard error = 14.3.*2546

*I can also find the p value and I'm guessing my p value is probably be tiny.*2564

*Here 14.3 is really small. *2573

*My p value is going to be t dist because I want my probability. *2577

*I put in my t, my degrees of freedom which is 7, and I have a two-tailed hypotheses.*2586

*That is going to be 2 × 10*^{-6}.2593

*Imagine .000002 given this tiny p value much smaller than .05 we should say at step 5 reject the null.*2610

*We had predicted that we would reject the null because the CI, the confidence interval did include 0.*2630

*Good job confidence interval and hypothesis testing working together.*2636

*Example 2, see the download again, this data set shows the average salary earned by first-year college graduates.*2641

*Graduated at the bottom or top 15% of their class for random sample of 10 colleges ranked in the top 100 public colleges in the US.*2650

*Is there a significant difference in earnings that is unlikely to have occurred by chance alone?*2661

*We want to know is there a difference between these top 15% folks and the bottom 15% folks.*2667

*They are linked to having graduated from the same college. *2674

*We would not necessarily want to compare people from the top 15% of one college that might be really good to one *2678

*to the bottom percentage of people from a college that might be not as great.*2687

*We would really want from the same college does not matter if you are in the top 15 or bottom 15%. *2693

*If you go to example 2, you will see these randomly selected colleges and the earnings in dollars per year, salary per year for the bottom 15%, as well as the top 15%.*2699

*Because it is a paired sample what we want to do is start off with d or set up d.*2718

*What is the difference between bottom and top?*2724

*We are going to get probably a whole bunch of negative numbers assuming that top earners earn more than bottom.*2729

*Indeed we do, we have a bunch of negative numbers.*2738

*If you wanted to turn these negatives into positives, you just have to remember *2740

*which one you decided as x and which one you decided to be y.*2745

*I will call this one x and I will call this one y.*2750

*It will help me remember which one I subtracted from which.*2759

*I am going to reverse all of these and it is just going to give me the positive versions of this.*2764

*Here is my d.*2771

*Let us go ahead and start with hypothesis testing.*2773

*This part I will do by hand.*2777

*Step 1, the null hypothesis says something that the top 15% folks and the bottom 15% folks are the same.*2783

*Their difference is going to be 0. *2796

*The mu sub d bar should be 0 but the alternative is that they are different.*2800

*We are neutral as to how they are different.*2807

*We do not know whether one earns more than the other.*2811

*Whether they are top earns more than bottom or bottom earns more than the top.*2813

*We can use our common sense to predict that the top ranking folks might earn more, but right now we are neutral.*2818

*Step 2, is our alpha level .05 or significance level. *2827

*Let us say two details.*2834

*Step 3, drawing the SDOD, the mean differences and here we will put 0.*2837

*And let us figure out the standard error.*2850

*The standard error here would be s sub d bar and that would be the standard deviation of d / √(n ) sub d.*2857

*We also want to figure out the degrees of freedom so that is going to be n sub b -1 and we also want to find out the t.*2871

*These are all things you can do in Excel.*2881

*Step 3, standard error is going to be s sub d bar and that will be s sub d ÷ √n sub d.*2884

*That will be the standard deviation of our sample of d ÷ the square root of how many there are and there is 10.*2903

*Here is our standard error. *2923

*What is our degrees of freedom?*2926

*That is going to be 10-1 =9. *2930

*What is our critical t?*2935

*We know it is a critical t because we are still in step 3 the decision stage.*2939

*We are just setting up our boundaries.*2944

*That is going to be t inverse because we already know the probability .05 two-tailed,*2947

*degrees of freedom being 9 and we get 2.26.*2953

*It is + or -2.26 those are our boundaries of t.*2959

*Step 4, this will say what is our sample t?*2966

*And that is going to be our d bar – mu / standard error/*2973

*I will write step 4 here and so I need to find t which is d bar – mu/ standard error.*2981

*I need to find the bar for sure and standard error.*2994

*My d bar is the average of all my differences and that is about $12,000 - $13,000 a year.*3000

*That is just right after college. *3016

*I need to find the d bar - 0 ÷ the standard error to give me my sample t.*3018

*That is the difference between sample t and critical t.*3033

*8.05 is actually the average of the differences.*3041

*The top 15% are on average earning $13,000 more than the bottom 15%.*3056

*The sample t gives us how far that differences from 0 in terms of standard error.*3065

*We know that is way more extreme than 2.26.*3075

*Let us find the p value.*3080

*We put it in t dist because we want to know the probability.*3083

*Put in our t, degrees of freedom, and we have a two-tailed hypotheses.*3087

*That would be 2 × 10*^{-5}.3094

*Our p value = 2 × 10*^{-5} which is a very tiny, tiny number, much smaller than the alpha.3100

*We would reject the null hypotheses.*3113

*Is there a significant difference in earnings that is unlikely to have occurred by chance alone?*3118

*There is always going to be a difference in earnings between these two groups of people, the top 15 and the bottom 15%.*3125

*Is this difference greater than would be expected by chance?*3131

*Yes it is because we are rejecting the model that they are equal to each other. *3135

*Example 3, in fitting hearing aids to individuals, researchers wanted to examine whether *3141

*there was a difference between hearing words in silence or in the presence of background noise.*3151

*Two equally difficult wordless are randomly presented to each person. *3156

*One less than silence and the other with white noise in a random order for each person.*3160

*This means that some people get silence than noise, other people get noise and silence.*3166

*Are the hearing aid equally effective in silence or with background noise?*3171

*First conduct the t test assuming that these are independent samples then conduct the t test assuming that these are paired samples.*3178

*Which is more powerful?*3185

*The independent sample t-test or paired samples t test?*3188

*We need to figure out what it means by more powerful.*3192

*I need some scratch paper here because the problem was so long I am just going to divide the space in half.*3196

*This top part I am going to use for assuming independent samples.*3205

*They are not actually independent samples, but I want you to see the difference between doing them as independent sample and doing them as paired samples doing this hypothesis testing as paired samples.*3211

*Step 1, the hypothesis, the null hypothesis is that if I get these sample *3224

*and they are independent this difference of means on average is going to be 0. *3231

*The mu sub x bar - y bar is going to = 0. *3240

*The alternative hypothesis is that the mu sub x bar - y bar does not equal to 0.*3244

*Here I am going to put alpha =.05, two-tailed.*3252

*I am going to draw myself an SDOD.*3261

*Just to let you know it is the differences of means.*3268

*Here we know that this is going to be 0 and we probably should find out the standard error.*3273

*The standard error of this difference of means is going to be the square roots of the variance of x bar + the variance of y bar.*3283

*I am going to write this out to be s sub x*^{2}/ n sub x + s sub y^{2} /n sub y.3303

*The variance of x and x bar, the variance of x /n, the variance of y/n.*3315

*We will probably need to find the degrees of freedom and that is going to be n sub x – 1 + n sub y -1.*3321

*Finally we will probably need to know the critical t but I will put that up here.*3340

*Let us look at this data, go to example 3. *3345

*Click on example 3 and take a look at this data.*3353

*Let us assume independent samples.*3356

*Here we are going to assume that this silence is just one group of scores and *3359

*this background noise is another group of scores and they are not paired.*3368

*They are actually paired.*3372

*This belongs to subject one, these 2 belongs to subject 3, this belongs to subject 5.*3374

*Here is the list order, it is A, B.*3380

*We get A list first then list B and here is the noise order.*3384

*They get it silent first then noisy.*3388

*This guy gets noisy first then silent.*3390

*All these orders are randomly assigned and the noise orders are randomly assigned as well.*3393

*For this exercise, we are going to assume we do not have any of this stuff.*3406

*We are going to assume this is gone and that this just a bunch of scores from one group of subjects *3412

*that listen to a list of words in silence and another group of subjects that listen to list of words in background noise. *3418

*We do the independent samples t test and we start with step 3. *3427

*We know we need to find the standard error, which is going to be the square root of the variance of x ÷ n(x) + the variance of y / n sub y.*3433

*All that added together and a square root.*3473

*We need to find the variance of x.*3476

*We need to find n sub x.*3479

*We also need to find the variance of y and n sub y before we can find standard error. *3481

*Variance is pretty easy.*3488

*We will just call silence x and the count of this is 24.*3491

*The count for y is going to be the same, but what is the variance of y?*3504

*The variance of y slightly different.*3513

*In order to find this guy, the standard error, we are going to put in square root of the variance of x ÷ 24 + the variance of y ÷ 24.*3521

*We get a standard error of 2.26 and standard error gives it just in terms of number of words accurately heard. *3547

*We also need to find the degrees of freedom.*3564

*In order to find degrees of freedom, we need the degrees of freedom for x + degrees of freedom for y.*3567

*The degrees of freedom for x is just going to be 24 - 1 and the degrees of freedom for y is also going to be 24 – 1.*3574

*The new degrees of freedom is 23 + 23 = 46.*3586

*Once we have that we can find our critical t.*3593

*Our critical t, we know that alpha is .05 so we are going to put in t in and *3606

*put in our two-tailed probability and the degrees of freedom 46.*3615

*We get a critical t of + or -2.01.*3620

*Our critical t is + or -2.01.*3625

*I will just leave that stuff on the Excel file.*3631

*Given all this now let us deal with the sample.*3636

*When we find the sample t what we are doing is finding the difference in means and then find the difference *3641

*between that difference and our expected difference 0 and divide all of that by standard error to find how many standard errors away we are.*3653

*Here I will put step 4, sample t.*3666

*In order to find sample t we need to find x bar - y bar - mu and all of that ÷ standard error.*3672

*Thankfully, we have a bunch of those things available to us quite easily.*3697

*We have x bar, we can get y bar, we can get standard error.*3701

*Let us find x bar, the average number of words heard accurately in silence and that is about 33 words.*3708

*The average number of words heard correctly with background noise, and that is 29 words.*3723

*Is the difference of about 4 words big enough to be statistically different?*3732

*We would take this - this and we know mu = 0 so I am going to ignore that / standard error found up here.*3741

*That would give us 1.75.*3754

*1.75 that is more extreme than + or -2.01.*3758

*1.75 we will actually say do not reject.*3765

*We should find the p value too.*3769

*This p value should be greater than .05.*3773

*We will put in t dist then our sample t, degrees of freedom which is 46 and we want a two tailed and we get .09.*3777

*.09 is greater than .05.*3790

*Step 5, fail to reject.*3797

*Now that we have all that, we want to know is it more sensitive?*3809

*Can we detect the statistical difference better if we used paired examples?*3821

*Let us start.*3829

*Here we would say p =.09 and 5 is failed to reject.*3831

*It is not outside of our rejection zone, it is inside our fail to reject zone.*3846

*Let us talk about the null hypotheses here.*3855

*What we are going to do is find the differences first then the mean of those differences.*3860

*We are saying if they are indeed not that different from each other that mean different should be 0. *3866

*The alternative is that the mean difference is not equal to 0.*3871

*Once again alpha = .05 two tailed and now we will draw our SDOD bar which means *3877

*it is a standard sampling distribution of means mean made of differences.*3892

*Here we want to put 0.*3909

*We probably also want to figure out standard error somewhere along the line, *3919

*which is going to be s sub d bar which is s sub d ÷ √n sub d.*3924

*We probably also want to find the degrees of freedom, which is going to be n sub d -1.*3933

*We probably also want to find the critical t.*3941

*Let us find out that.*3945

*Here I will start my paired samples section.*3949

*I will also start with step 3. *3955

*Let me move all of these over here.*3957

*Let us start here with step 3*3965

*Let us find standard error and that is going to be s sub d not d bar ÷ √n sub d.*3970

*We can find s sub d very easily and we could also find n sub d.*3986

*First we need to create a column of d.*3994

*I will find the standard deviation of the d but I realized that I do not have any d.*4002

*The d look something like this silence - background noise.*4008

*This is how many more words, they are able to hear accurately in silence and background noise.*4020

*Here we see that some people hear a lot of words better in silence.*4026

*Some people here words better with a little bit of background noise. *4032

*Some people are exactly the same.*4035

*We could find a standard deviation of all these differences.*4037

*We could also find the mean of them*4045

*The n of them will be the same as 24 because there are 24 people that came from.*4053

*There is 24 differences.*4062

*We could find out standard error. *4065

*Standard deviation of d ÷ √24.*4070

*That is standard error, notice that is quite different from finding a standard error of independent samples.*4076

*Let us find degrees of freedom for d and that is going to be n sub d -1 and that is 24 -1.*4086

*Our critical t should be t inverse .05 two tailed=23 and we get 2.07.*4101

*So far it seems that our standard for how extreme it has to be is more far out.*4127

*That makes sense because the degrees of freedom is smaller than 46.*4134

*+ or -2.07.*4139

*Let us talk about our sample.*4152

*In order to find our sample t, we want to find the average of difference subtract from the hypothesized mu *4154

*and divide all of that by standard error to find out how many standard errors away our sample mean difference is.*4169

*We also want to find p value.*4179

*Here is step 4, our sample t would be d bar - mu ÷ standard error. *4181

*What is d bar and how would we find it?*4196

*Just use the average function and the average of our d like this d bar.*4205

*We can do d bar -0 / standard error= 2.97.*4211

*That is more extreme than 2.06.*4226

*Let us figure out why.*4233

*We might look at standard error, the standard error is much smaller and the steps are smaller.*4235

*How many steps we need to take to get all the way out to this d bar?*4251

*There is more of them than these the bigger steps. *4257

*These are almost twice as big.*4261

*These bigger steps, there is few of them that you need.*4263

*That is what the sample t I get is how many of these standard errors, *4267

*how many of these steps does it take to get all the way out to d bar or x bar – y bar?*4273

*We need almost 3 steps out.*4279

*What is our p value?*4282

*Our p value should be less than .05 that is going to be t dist.*4287

*Here is our t value I will put in our degrees of freedom and two tailed and its .007. *4293

*That certainly less than .05.*4303

*Step 5, here we reject whereas here we fail to reject.*4306

*Since there is this difference and we detected it with this one but not with this one, *4313

*we would say that this is the more sensitive test given that there is something to detect out there.*4321

*This is the difference if it does exist. *4328

*This one is a little coarser, there is a couple of reasons for that.*4331

*One of the reasons is because the standard error are usually larger than the standard error of differences. *4336

*Another issue is that x bar - y bar, the difference here if we look at x bar - y bar this difference is roughly around the same.*4343

*This difference is the same as this difference.*4362

*It is not that bad but it is that you are dividing by a smaller standard error here then you are here.*4366

*Here, the standard error is quite large. *4373

*The steps are quite large. *4375

*Here, the standard errors are small.*4376

*The steps are quite small.*4379

*It is because you are taking out some of the variation caused by having some people *4380

*just being able to hear a lot of words accurately all the time with noise.*4385

*Some people are very good at hearing anyway.*4392

*They might have over a low number of scores but with d bar you do not care about those individual differences.*4397

*You end up accounting for those by subtracting them out.*4405

*Here this is a more sensitive test.*4409

*Here we get p=.006 and we reject.*4412

*Which test is more sensitive?*4425

*Which test is able to detect the difference, if there is a difference?*4431

*Paired samples.*4435

*That principles are little more complicated to collect that data but it is worth it because it is a more sensitive test.*4436

*Thanks for using www.educator.com.*4448

*Hi and welcome to www.educator.com.*0000

*Today we are going to talk more in-depth about type 1 and type 2 errors.*0001

*If you want to know more about power and effect size it is good to go through this lesson *0006

*because it is going to help you understand some of the pictures that we are going to draw in the future. *0013

*Here is the roadmap for today.*0017

*We need to know about these type 1 and type 2 errors, but we also need to know when we make those errors in relationship to hypothesis testing.*0021

*So far we only used t test as our hypothesis test.*0033

*We have shown these errors and their relationship to hypothesis testing before as a box, but frequently in hypothesis testing we draw distributions.*0037

*The SDOM to be more specific.*0048

*What I want to show you how the errors fit on this distribution picture.*0051

*We are going to show you how the box and the distributions fit together because these two things actually relationship to each other. *0058

*They refer to the same concept. *0065

*There are just 2 different ways of showing you that same concept.*0067

*We go through hypothesis testing, but in the real world there is some reality that either the null hypotheses is just true or the null hypothesis is false.*0071

*Although we do not know this reality, all we know is the result of our hypothesis testing.*0086

*There are two kinds of ways we can make errors.*0092

*We can make an incorrect decision by false alarming.*0095

*We reject the null, but we should not have rejected the null.*0099

*That is called the false alarm or a type 1 error. *0106

*I used to get confused between which one is type 1 and type 2, these are arbitrate. *0110

*I like to think of this as the more serious error when you successfully reject the null hypothesis that is a more extreme thing that you do.*0116

*This is actually more dangerous than this miss.*0127

*That is not much of an error but actually false alarming.*0131

*That is how I remember the number 1 error you should look out for.*0136

*The type 1 error is often also called the likelihood of false alarming.*0142

*The probability of false alarming and that is referred to as alpha.*0151

*If the reality that we do not know is that this null hypothesis is true we have a probability of false alarming with the rate of alpha.*0157

*We have the probability of failing to reject when we should have rejected, a correct failure your probability is 1-alpha.*0171

*These two things add up to 1.*0186

*The probability of false alarming + the probability of making a correct failure =1.*0189

*On the flipside, let us say that null hypothesis is false that is not a true picture or model of the world. *0199

*Then we really should have reject it.*0210

*It is not true, we should reject it, that would be a correct decision and that is called the hit where we are rejecting the null when we should have rejected it.*0213

*That gives us the probability of hits.*0226

*We could be incorrect and fail to reject when we should have rejected that is also another incorrect decision.*0230

*That is the type 2 error.*0242

*It is a miss and the probability of miss is given as beta.*0244

*Beta + 1 –beta = 1. *0248

*The probability of misses + probability of hits =1.*0254

*In which of these boxes is the sample statistic statistically significant?*0262

*In which of these boxes is our p value less than .05 or whatever our alpha level is.*0274

*Let us think about that.*0280

*When we reject the null hypothesis that means our test statistic in this case t is extreme.*0282

*Our p value is significant and remember we mean significant as it stands out. *0294

*It is very weird. *0304

*In this case, these two quadrants up here is what we should worry about.*0307

*This is the decision we need to worry about when we reject the null hypothesis.*0315

*The other possibility is that when we reject the null hypotheses and our p is significant we made a correct decision.*0322

*These are our two choices if we know that p is less than alpha or if our test statistic is extreme.*0334

*Here p is not significant. *0343

*It is not too weird and because of that we will fail to reject and we can be correct in failing to reject *0349

*or when we fail to reject we could be wrong by making a type 2 error.*0358

*Here is what I want you to know. *0363

*Let us say we carry out hypothesis testing and I think I have a really low p value.*0365

*I am going to reject my null hypotheses.*0372

*Which error am I likely to make, a false alarm or a missed?*0375

*Since I rejected my null, the only error I can possibly make is this one where I reject the null and get wrong.*0381

*Let us say I go through my hypothesis testing and I get p=.4.*0397

*Let us say I do not reject my null.*0405

*What mistake or what error could I have possibly made?*0408

*The only error I can make is the missed error.*0411

*Here I fail to reject and I could be wrong in doing it.*0414

*Let us talk about distributions and how errors fit in here.*0418

*We have a one sample t test we set up some null population.*0426

*This is our null hypotheses population and our hypothesized mu might be 230.*0431

*We do not know whether our sample is part of this or it is part of some other population, not the null population.*0443

*We can hypothesize maybe it comes from some other population like this one.*0454

*When we set our alpha levels and create critical t and zones of rejection and all of that stuff what we are doing is recreating the line.*0459

*If our sample t is outside here then we are going to reject the null.*0476

*So far we have only colored in this part, but we really mean this part as well as all of this part.*0492

*That is our reject the null zone, this entire area. *0508

*In order to find out whether we should reject the null or not we also need to look past the raw score.*0514

*We need to look past the raw score and we need to look at it in terms of the critical t.*0528

*The critical t might be whatever like -2. Something .*0536

*We need to find out this t value and so I am just going to make one.*0544

*Let us say this t value is 5.5 and if our t value is sufficiently extreme then we reject are null hypothesis.*0549

*This would be our critical t and this is our sample x bar, but this is our sample t.*0560

*And that is how it looks out here.*0574

*Our possibility of making an error is this little gray spot that I have colored in red.*0577

*Just in case my sample really does come from these areas, I should not have rejected the null.*0587

*If it happens by chance rule 50 heads in a row it is very unlikely but it is still possible.*0596

*It is still possible that I got this x bar even though this is the true population distribution.*0613

*This is my possibility of making a type 1 error.*0621

*We actually have to add this side up to this side type 1 error.*0628

*We know that this is alpha=.05.*0640

*This part is 1 – alpha which is .95 and that is our possibility of not rejecting given that the null hypothesis is true.*0646

*That is the example of one sample hypothesis testing.*0660

*This is the same picture as before.*0666

*I just written it more neatly for you by typing it out and you can think of this test statistic as just t.*0669

*I have just written the generic word test statistic to think of this as critical t and sample t.*0676

*Here is the important thing to realize.*0682

*This gray distribution here represents an SDOM and that is why this is mu sub x bar and there is also an x bar here as a sample. *0685

*This SDOM actually represents the probability where the null hypothesis is true and that probability equal to 1.*0696

*Remember we talked about that before when we said the area underneath the normal distribution equal to 1.*0706

*This represents the possibility that this may not be true and that there exists some other population that our sample really came from.*0713

*We do not just know what that population is.*0727

*That is the probability that the null hypothesis is false.*0731

*That normal distribution also has an area =1.*0736

*What we can additionally find out is when we create the zones of rejection and we say anything outside of this critical t reject it.*0744

*We color in this area here.*0759

*What we are saying is this is the probability of rejecting given that the null hypothesis is true. *0761

*This is the area where we fail to reject.*0777

*This probability right here represents the conditional probability of failing to reject given that h knot or null hypothesis is true.*0784

*And that equals 1 – alpha because this one equal alpha.*0807

*Those are the important things to remember. *0815

*These are all conditional probabilities as we learned about previously in probability lessons.*0819

*Let us talk about a two sample t test.*0826

*The idea behind the two sample t test is almost exactly the same except there are just a couple of exceptions now.*0830

*Instead of a raw score we have difference of scores and we still have a test statistic.*0838

*Here our mean hypothesized difference between our non college sample and our college sample is going to be 0 because that means they are the same.*0846

*Remember, these are SDOD (Sampling distributions of differences of means).*0862

*This is 0 and this might be our actual sample difference x bar – y bar, the actual difference between the samples.*0875

*Same thing down here, we have this as our critical test statistic and this is our sample t.*0887

*We want to know whether our sample t is way far out, more extreme than our critical t.*0902

*Here this represents the probability that the null hypothesis, that there is no difference is true and that is =1.*0910

*Same thing here, the probability that the null hypothesis is false and actually there some other distribution we just do not know what that is.*0923

*We will draw it like a ghost with blue.*0933

*It is important to know that this mu is mu sub x bar - y bar because we are talking about SDOD.*0936

*That is why it is a difference of means. *0946

*Once we know this, now what we need to do is figure out what these probabilities mean. *0950

*Here, let me draw the cut off again, here we have our rejection zone and our fail to reject zone.*0958

*Once again we can find those conditional probabilities. *0977

*What is the probability of rejecting given this thing is true, inside of this space where the null hypothesis is true?*0981

*What is the probability of failing to reject given that the null hypothesis is true?*0992

*That is the conditions that we are working under.*0999

*It is still the same. *1005

*Here we see alpha and here we see 1 – alpha.*1008

*Ideally when we have these differences between distributions what we really would like is that *1018

*there was very little overlap between these two distributions. *1027

*The null distribution and the like real one that we do not know anything about.*1031

*It will be nice if there was very little overlap.*1036

*But in real life, there is usually a lot of overlap.*1038

*The real world is noisy and the real population might be very, very different. *1043

*The real population might be very similar to the null population.*1055

*If that is the case, there is some overlap between their distributions.*1071

*There are some chances that we might get a score over here and it could be part of the real population or part of the null population.*1077

*If this is the case and we need to understand these conditional probabilities in anyway.*1086

*Get ready here is the deal. *1098

*Instead of writing real population, I am going to say not null population because we do not know what it is.*1100

*It is just not the null population.*1112

*I am going to take this picture, this great curve and I will draw up here in two ways.*1115

*I am going to split it up into two parts. *1121

*One part is going to be this blue part, this fail to reject region and that is that whole part.*1123

*Here I am also going to draw the red part.*1144

*I just draw it separated from each other so that you can see.*1147

*Here we have this little part and that is red and it is red because we have rejected it.*1158

*This is the case where we are actually wrong.*1167

*This is the case where we are actually right.*1171

*Here we are wrong because we rejected the null hypotheses that we should not have rejected.*1174

*Here we are correct, because we fail to reject and truly we should not have rejected it.*1180

*Now that is the case if the null hypothesis population is true.*1185

*What happens in a case where it is not true?*1193

*The null hypothesis is false.*1200

*What happens here?*1203

*Here I am going to draw a different looking picture because I'm going to draw this curve but this curve split up.*1206

*Here I am going to split this curve up like this. *1218

*On this side of the line I am going to draw this little section and draw just this little section.*1227

*That part of it I have failed to reject.*1253

*That is wrong so I am going to color it in red because we should have rejected it but we fail to reject it.*1257

*On the other side, I am going to try the other part of this curve.*1274

*It is this part and here I am going to color that in blue because although we rejected it we should have rejected it.*1279

*Here we rejected the null hypothesis and you are right we should have rejected it because we are in this new unknown population.*1292

*You should have rejected it.*1308

*Let us look at the places where we are correct.*1310

*We are correct here and this is called a correct failure.*1314

*Here we are also correct and this is called a hit.*1319

*Here we are incorrect and that is called a false alarm.*1331

*Here we are also incorrect and this is called a miss. *1344

*It is a miss because we have failed to reject it.*1352

*We failed to hit the target when we should have hit the target.*1357

*Given that, let us see how the distributions and the box go together.*1361

*The false alarm is really that place.*1369

*Remember when the hypothesis is true I am going to draw it in black.*1373

*The correct decision is going to be this whole section where we fail to reject, but that is okay we are in this fail to reject zone.*1378

*You are good to go. *1393

*Here is the other part of this part.*1395

*Here this is an error because we have rejected when we should not have rejected because it is actually true.*1401

*This is our false alarm. *1416

*Now, in the case of a correct decision where you actually hit it, this means you rejected it and it is good *1418

*that you rejected it because actually a different population is true, not this null population.*1430

*That is going to be the area where you reject, all rejections are going one on the right side of this line.*1438

*You should have rejected it because you are in a different population.*1454

*You are not in the null population.*1461

*This is a good thing for you.*1462

*You should have rejected it.*1466

*The other part of that, the other piece of that is down here.*1469

*It is this little piece down here.*1474

*Here it is incorrect, because although you are part of a different population, not the null population, you did not reject it.*1477

*You fail to reject.*1491

*I want you to notice something here.*1493

*All the fail to reject are always on this side of the line because these are values that are less extreme than the mean.*1500

*And the rejection ones are all in this side of the line. *1508

*I could also drawn it two tailed and also showing you the side but I'm showing you one tailed.*1511

*It is all outside of the line, on the outer boundaries of this line, more extreme than the hypothesized mean.*1517

*This is less extreme than the hypothesized mean.*1524

*My hypothesized mean is somewhere here, less extreme than that.*1527

*It is relative to the hypothesized mean.*1533

*That is how these four pictures fit together.*1539

*When you see those two distributions drawn, do not get confused you already know it. *1543

*You just have to break it apart in slightly different ways.*1548

*Let us go on to some examples.*1553

*On the basis of results from a large sample of students from a university, a professor reports the mean high from my sample is not significantly below 60.*1556

*That means he did not reject.*1573

*This is fail to reject.*1576

*If he said significantly that would be rejecting the null.*1581

*Which type of error will this professor worry about?*1586

*He failed to reject, that is important to know.*1590

*What is the only error you can make if you fail to reject?*1593

*Well if you fail to reject, but you should have rejected it, the null hypotheses is false, what kind of error is that?*1596

*That is a missed and a type 2 error.*1617

*The error rates are given by alpha and beta and this is actually beta so these are wrong. *1624

*These are both correct rates instead of the error rates and this is nonsense having a non significant results are all error statistically.*1631

*It is never the case.*1642

*You are damned if you do and damned if you do not.*1643

*There is always a way you can make an error either type 1 or type 2.*1645

*Example 2, a researcher worries about trying incorrect conclusion.*1649

*The researcher plan to select a sample of size 20 and to use the .01 level of significance.*1655

*Here alpha is .01.*1662

*In a two tailed test of the null hypothesis the critical t should be + or - because it is a two tailed test.*1664

*It is + or -2.86. *1676

*If he obtains the t of 2.8 which type of error would he be worried about and why?*1681

*Well, you definitely know that he is not going to reject.*1695

*Fail to reject because this is less extreme than this.*1704

*This is less extreme so he fail to reject.*1717

*The only error you can have when you fail to reject is if you fail to reject given the null hypothesis is false.*1722

*What kind of error is that?*1729

*That is a missed or type 2.*1733

*What if he obtains a t of 2.869 which type of error would he be worried about?*1744

*That is more extreme than this.*1752

*In this case he would reject the null.*1754

*When is he wrong when he rejects?*1757

*When he should have not rejected it because the null hypothesis is actually true.*1760

*What kind of error is that?*1765

*That is a false alarm or type 1 error.*1768

*Example 3, what is the danger of the type 1 error?*1776

*This is a more conceptual question. *1782

*The danger is mistakenly concluding that there is no significant difference between the obtained mean and the hypothetical population mean. *1785

*When you make a type 1 error you have rejected the null but null hypothesis is true.*1794

*Mistakenly concluding that there is no significant difference but that is not true *1808

*because you concluded that there is a significant difference that is why you rejected the null.*1814

*Mistakenly concluding that there is a significant difference between the obtained mean and the hypothetical population mean.*1818

*That is true.*1826

*You mistakenly rejected the null and said there is a significant difference but you should not have done that.*1829

*Mistakenly being alarmed about a hypothesis when you should become.*1838

*That is non sense.*1843

*Mistakenly calculating the wrong test score.*1844

*These errors are not errors that you can actually avoid.*1847

*These are not errors because we were sloppy. *1851

*These are errors that are made because we do not know the real nature of the world. *1854

*This is actually not what we are talking about when we are talking about type 1 or 2 errors.*1860

*Mistakenly choosing the wrong population standard deviation to calculate standard error, that is not it either.*1865

*These two are just regular old mistakes or errors in calculation.*1872

*They are not type 1 and 2 errors of hypothesis testing.*1878

*That is it for type 1 and 2 errors.*1881

*Thank you for using www.educator.com.*1885

*Hi, welcome to educator.com. *0000

*We are going to talk about effect size and power. *0002

*So effect size and power, 2 things you need to think about whenever you do hypothesis testing. *0005

*So first effect size. *0011

*We are going to talk about what effect sizes is by contrasting it to the T statistic. *0013

*They actually have a lot in common but there is just one subtle difference that makes a huge difference. *0019

*Then we are going to talk about the rules of effect size and why we need effect size. *0026

*Then we are going to talk about power. *0032

*What is it, why do we need it, and how do all these different things affect power for instance sample size, *0035

*effect size, variability in alpha, the significance level. *0042

*So first things first, just a review of what the sample T really means. *0048

*So a lot of times people just memorize the T formula, it is you know the X bar minus mu over standard error but think about what this actually means. *0056

*So T equals X bar minus mu over the standard error. *0070

*And all right that is S sub X bar. *0076

*What this will end up giving you is this distance so the distance between your sample and your hypothesized mu. *0079

*And when you divided by standard error you get how many standard errors you need in order to get from bar to your mu. *0087

*So you get distance in terms of standard error. *0097

*So distance in terms of standard error. *0102

*And you want to think of in terms sort of like instead of using like feet or inches or number of friends, we get distance in the unit of standard error. *0111

*So whatever your standard error is for instance here that looks about right, because this is the normal *0123

*distribution that should be about 68% so that is the standard error. *0132

*Your T is how many of these you need in order to get to T. *0139

*So this might be like a T of 3 1/2, 3 1/2 standard errors away gets you from mu to your sample difference and so this is the case of the two sample t-test. *0146

*So independent samples are paired samples where we know the mu is zero. *0164

*So this is sort of the concept behind the T statistic. *0169

*Now here is the problem with this T statistic. *0175

*It is actually pretty sensitive to N. *0181

*So let us say you have a difference that is very, that is going to stay the same so a difference between, you know let us say 10 and 0. *0185

*So we have not that difference. *0199

*If you have a very very large N then your S becomes a lot skinnier. *0202

*And because of that your standard error is also going to shrink so the standard error shrinks as N shrinks. *0212

*And because of that, even though we have not changed anything about this mean, about the X bar or mu, *0226

*by shrinking our standard error we made our T quite large. *0237

*So we made our T like all of a sudden were 6 standard or errors away but I really have not changed the picture. *0244

*So that is actually a problem that T is so highly affected by N. *0252

*The problem with that is that you could artificially make a difference between means, look statistically significant by having a very large N. *0258

*So we need something that tells us this distance that is less affected by N and that is after effects size comes in. *0268

*So in effect size what we are doing is we want to know the distance in terms of something that is not so affected by N. *0278

*And in fact we are going to use the population standard deviation because let us think about T. *0288

*So that is X bar minus mu over standard error. *0295

*So this contrast that to looking at the distance in terms of the standard deviation of the population, what would that look like. *0303

*Well, we could actually derive the formula ourselves. *0307

*We want that distance in terms of you know number of inches or number problem correct or whatever the *0321

*raw score is over instead the standard error we would just use S or if you had it you would use Sigma. *0328

*So you could think of this as the estimated Sigma and this is like the real deal Sigma. *0341

*And that is what effect size is and effect size is often symbolized by the letters D and G. *0349

*D is reserved for when you have when you have Sigma, G is used for when you use S. *0360

*Now let us talk about the roles of effect size. *0367

*The nice thing about effect size is that the N does not matter as much whether you have a small sample or large sample the effect size stays similar. *0373

*In test statistics suggest T or Z, the N matters quite a bit and let us think again about why. *0384

*So the T statistic I have been writing at so far as over standard error but let us think about what standard error is. *0396

*Standard error is S divided by the square root of N, now as N gets bigger and bigger so let us think about N getting bigger. *0406

*This whole thing in the denominator, this whole idea this whole thing becomes smaller and smaller. *0417

*And when you divide a positive or negative or positive, if you divide some distance by a small number then *0430

*you end up getting a more extreme value, more extreme. *0441

*So by more extreme I mean way more positive, more positive ,more on the positive side or way more negative. *0448

*So the T statistic is very very sensitive to N so is the Z because Z the only difference is instead of S we use Sigma. *0463

*And so the same logic applies but for effect size T and G we do not divide by square root of N so in that way N does not really have as much. *0474

*Okay so one thing to remember is if you know Sigma use covens D, if you need to estimate the standard deviation from the sample S, you want to use hedges G. *0488

*Okay so now you know what effect size is and it is nice that it is not as affected by N but why do we need it? *0500

*Well effect size is what we use, the statistically used to interpret practical significant so for instance we 0839.8 might have some sort of very small difference between group 1 and group 2 so with the males and females *0510

*on some game or task, there is a very tiny difference like you know let us just say males are ahead by .0001 points. *0527

*And practically, it sort if does not matter but if you have a large enough effect size if you have a large *0540

*enough N you could get a small enough standard error that you can make that tiny difference seem like a *0549

*big deal and you can imagine that would be sort of odd situation. *0556

*We take a difference that sort of does not matter but then make a big deal out of it because of some fancy statistics we did. *0565

*Well, that effect size is not going to be affected by N and so that going to give you more straightforward *0573

*measure of is this difference big enough for sort of just care about. *0580

*It is not going to tell you whether it was significant or not based on hypothesis testing but it can give you*0584

*the idea of practical significant and here were using the modern term for significant as in important. *0592

*It will tell you a bit practical importance not statistical outlier nests, that is how it is telling you, it is talking*0601

*about just regular old practical importance and the way you can think about this is just thinking about it as is this different worth noticing. *0614

*Is that worth even doing statistics on? *0623

*The thing about hypothesis testing is that it could be deceiving, a very large sample size can lead to a *0625

*statistically significant one of these outlier differences that we really do not care about that just has no practical significant. *0632

*So here although we have been trying to talk about this again and again trying to sort of clarify that *0641

*statistically significant does not mean important it just means it lies outside of our expectation. *0648

*It is important to realize once again that statistical significance does not equal practical significant. *0656

*This is sort of talking about how important something is and this is just sort of saying, does it stand out? *0663

*Does our X bar our sample actually stand out? *0672

*Okay now let us move on to power. *0679

*What is power? *0684

*Well, how we really needs to go back to our understanding of the two types of errors. *0685

*Remember in hypothesis testing we can make an error in two different ways. *0691

*One is the false alarm error and we set that false alarm error rate by Alpha and the other kind of error is *0695

*this incorrect decision that we can make called the miss. *0704

*A miss is when we fail to reject the hypothesis, that the null hypothesis but we should, we really should. *0708

*And that is signified by beta, by the term beta. *0717

*Now when the null hypothesis is true then we can know if we already set our probability of making *0725

*incorrect decision, just like subtraction we can figure out our probability of making a correct decision so if 1225.5 our probability is .05 in making incorrect decision, the other possibility is that we may correct decision 95%, 1-.05. *0739

*In the same way when the null hypothesis is actually false we could figure out our probability of actually *0756

*making a correct decision by just subtracting our probability of making incorrect decision from one. *0764

*So this would be one minus beta. *0772

*In that way these two decisions that we make they add up to a probability of one and this 2 decisions that we can make add up to probability of one. *0775

*But in reality only one of these worlds is true that is why they both have a probability of 1. *0787

*We just have no idea whether this one is true or this one is true and anyone can never really say but that is the philosophical question. *0794

*So given this picture, power resides here and this quadrant is what we think as power. *0802

*Now power is just the idea given that the world is actually false, that this world we live in pretend we *0811

*ignore this part right so I am just, just ignore this entire world, given that this null hypothesis is false, what*0824

*is our probability of actually rejecting the null hypothesis and that is what we call power. *0835

*So think of this as the probability of rejecting null when the null is false. *0843

*So why do we need power, why do we need 1 – beta? *0855

*Well, here it is going to come back, those concepts come right back. *0864

*Remember the idea that you know sometimes we wanted to detect some sort of disease right and we *0873

*might give a test like for instance we want to know whether someone has HIV and so we give them a blood test to figure out, do they have HIV. *0879

*Now this test are not perfect and so there is some chance that they will be able to detect the disease and some chance that will make a mistake. *0888

*There is two ways that there is two ways of thinking about this prediction. *0897

*One is what we call a positive predictive, value we could think about what is the probability that someone has the disease for instance HIV given that they test positive? *0903

*Well this will help us know what is the chance that they actually have the disease once we know their test score. *0916

*In this world we know their test scores and we want to know what is the probability that they have the disease. *0926

*On the other hand we have what is called sensitivity. *0932

*Sensitivity thinks about the world in a slightly foot way. *0936

*Given that this person has the disease, has whatever disease such as HIV, one is the probability that they will actually test positive. *0940

*And said that at these two actually give us very different world. *0950

*In one world the given is that they have a positive test and what is the probability that they have the disease versus no decease. *0954

*In this scenario the given is very different. *0967

*The given is that they actually have the disease. *0973

*Given that what is the probability that they will test positive versus negative? *0976

*And so they are looking at this or they are looking at this. *0983

*Now power is basically the probability of getting a hit, the probability of rejecting that null hypothesis given 1637.9 that the null hypothesis is actually false so it is actually wrong. *0988

*Is this more like PPV, positive predictive value? *1004

*Or is it more like sensitivity. *1010

*Well let us think about this. *1012

*In this world there is this reality that the given reality is that this is false. *1014

*We need to reject it. *1022

*What is the probability that will actually be rejected? *1027

* So reject or fail to reject. *1032

*Well one way of thinking about this in a more, in the comparison is to consider, what is this thing that we *1040

*do not know in these two scenarios? *1052

*We do not really know if they actually have HIV. *1055

*We know their test we know that their test is either positive or negative and the test is uncertain but *1059

*whether they actually have HIV or not, that does not have uncertainty, it is just that we do not know what it is. *1065

*This is sort of like HIV in that way. *1074

*This is the reality so HIV is the reality and this, this is the test results. *1078

*This is also the reality and these are the results of hypothesis testing. *1088

*And so in that way this picture is much more like sensitivity. *1101

*And really when we apply the word sensitivity we see a whole new way of looking at power. *1107

*Power is the idea how sensitive is your hypothesis test when there really is something to detect, can it detect it? *1116

*When there really is HIV, can your test detect it? *1125

*When the null hypothesis really is false, can your test detect it? *1129

*That is the question that power is asking. *1136

*Okay if you calculate power is there nice little formula for? *1139

*Well power is more like the tables in the back of your book. *1145

*You cannot like calculate with like one simple straightforward formula. *1148

*There is actually more complex formula that does not both calculus but we can simulate power for a whole *1153

*bunch of different scenarios and those scenarios all depend on outline effect size and also variability in *1161

*sample size and because of that power is often found through simulation. *1169

*So I am not going to focus on calculating power, instead I am going to try to give you a conceptual understanding power. *1174

*Now often a desired level of power and sometimes you may be working with computer programs that might calculate power for you. *1187

*A different level of power that you want to shoot for is .8 or above but I want you to know how this power interact with all this things. *1195

*All of these things actually go into the calculation of power but I want you to know what is the conceptual level. *1206

*So how does Alpha or the significance level affect power, how does affect size, D or G affect power, how *1212

*does variability S or you know, S squared affect power and how to sample size affect power. *1224

*Okay so first thing is how does Alpha affect power? *1234

*Well here in this picture, I shown you 2 distribution. *1241

*You could think of this one is the null distribution and this one as the alternative distribution. *1247

*And noticed that both of these both of these distributions up here are exactly the same down here I just copied and pasted. *1254

*The only thing that is different is not their means or the actual distribution. *1263

*The only thing that is different is the cut off. *1276

*Since here, the cut off scores right here and this is the alpha, and hear the cutoff score has been moved sort of closer towards the population mean. *1279

*And now we have a huge Alpha. *1297

*So let us just assign some numbers here. *1301

*I am just guessing that maybe that looks like alpha equals .05 that something more used to see, but this looks like maybe Alpha equals let us say .15. *1304

*What happens when we increase our Alpha? *1317

*Our Alpha has gotten bigger, what happened, what happens to power? *1323

*Well it might be helpful to think about what Power might be? *1326

*In this picture, remember, power is the probability of rejecting the null hypothesis when the null hypothesis*1331

*is actually false and here we often reject when it is more extreme than the cutoff value when your X bar is *1344

*more extreme so these are the rejections of everything on this side. *1358

*All of this stuff is reject, reject the null. *1362

*And we want to look at the distribution where the null hypothesis is false a.k.a. the alternative hypothesis. *1369

*So really were looking at this big section right here so here this big section, that is power, one minus Beta*1380

*given that you could also figure out what beta is. *1396

*And beta is our error rate for misses. *1400

*When we fail to reject, fail to reject but the alternative hypothesis is true or the other way we could say it is the null hypothesis is false. *1406

*So what happens to power when Alpha becomes bigger? *1421

*Well, let us colour in power right here and it seems like there is more of this distribution that is been colored in than this. *1430

*So this part has been sort of added on, it used to be just this equals power but now we also have added on the section. *1440

*So as Alpha increases, power also increases. *1453

*And hopefully you can see that from this picture. *1464

*Now imagine moving Alpha out this way so decreasing Alpha. *1467

*If we decreased alpha then this power portion of the distribution that power part will become smaller so *1472

*the opposite, sort of the counterpoint to this is also true as Alpha decreases, the power also decreases. *1483

*But you might be asking yourself, then why cannot we just increase alpha so we could increase power right? *1496

*Well, remember what Alpha is, alpha is your false alarm rate. *1511

*So when you increase Alpha, you also increase your false alarm rate. *1516

*So at the same time if you increase your false alarm rate your increasing power. *1521

*And so this often is not a good way to increase power. *1526

*But you should still know, with the relationship is. *1533

*How about effect size, how does effect size affect power? *1538

*Well remember, effect size is really sort of a, you can think of it roughly as this distance between the X bar and the mu. *1544

*We are really looking at that distance in terms of standard deviation of the population. *1555

*How does effect size affect power? *1561

*Here I have drawn the same pictures, same cut off except I have moved this null, this alternative *1564

*distribution a little bit out to be a more extreme so that we now have a larger distance, larger distance. *1574

*And so this is a bigger effect size, bigger effect size so what happens when we increase the effect size and*1587

*we keep everything else constant, that the cut off, the null hypothesis, everything. *1600

*Well, let us colour in this, and colour in this. *1605

*Which of these two blue areas is larger. *1612

*Obviously this one. *1616

*This power is bigger than this power and it is because we have a larger effect size so another thing we have*1618

*learned is that larger effect size it leads to larger power so as you increase effect size you could increase power but here is the kicker. *1627

*Can you increase effect size? *1646

*Can you do anything about the effect size? *1649

*Is there anything you could do? *1651

*Not really. *1655

*Effect size is something that sort of out there in the data but you cannot actually do anything to make it *1657

*bigger but you should know that if you happen to have a larger effect size then you have more power than if your study of a small effect size. *1663

*Okay so how does variability and sample size affect power? *1674

*Now the reason I put these two things together is that remember, this distribution are S Toms, right? *1686

*And so the variability in a S Tom right is actually standard error and standard error is S divided by the square root of N. *1696

*So both variability and sample size will matter in power. *1711

*And so here I want to show you how. *1720

*Okay so here I have drawn the same means of the population of the S Toms and remember here we have *1723

*the null hypothesis and the alternative hypothesis distribution. *1733

*I have drawn the same pictures down here and I kept the same Alpha about .05. *1739

*So I had to move the cut off a little just so that I could color in .05 but something has changed and that is this. *1749

*This a lot skinnier than this is, that is less variability so that S Tom has decreased in variability. *1760

*So here standard error has decreased so we have sharper S Toms. *1772

*Still normally distributed, just sharper. *1786

*And so when we look at these skinnier distribution let us look at the consequences for power. *1790

*Here lets color in power and let us color in power right here and it also helps to see what beta is. *1798

*So here we have a quite a large beta and here we have a tiny beta. *1810

*And so that makes you realize that the one minus Beta appear the power here is larger than the one minus Beta down here. *1815

*This is smaller than the 1 – beta down here because remember were talking about proportions. *1832

*This whole thing add up to 1, now this might look smaller to you, the whole thing adds up to one. *1849

*If this is a really small proportion so let us do a number on it, that was less than .05. *1855

*Let us go on .02. *1861

*This looks bigger than .05 so let us go on .08. *1863

*Then 1 - Beta here would be 92% and 1 - Beta here would be 98% so this is a larger power than this. *1869

*So one thing we have seen is our standard error decreases so it is decreasing then power increases so this is what we call a negative relationship. *1879

*As one goes down the other goes up and vice versa as standard error increases as these distribution *1899

*become fatter and fatter then power will decrease overall the opposite way. *1908

*Now because we already know this about standard error we could actually say something about sample *1913

*size because sample size actually has the opposite it also has a negative relationship with standard error *1921

*and sample size go get bigger and bigger and bigger standard error gets smaller and smaller and smaller *1929

*and so sample size actually have a positive relationship with power so as sample size increases and *1936

*therefore standard error decreases, power increases. *1947

*And so we could figure that out just by reasoning through what standard error really mean. *1954

*Okay so how do we increase power because often times your power or sensitivity is really a good thing. *1965

*We want to be able to have experiments and studies that have a lot of power that would be a good hypothesis testing adventure to embark on. *1976

*How do we actually increase it. *1987

*Well can we just do this by changing Alpha? *1989

*Well the problem with this is that you get some consequences namely that falls alarms increase. *1994

*So if you increase power with this strategy you are also going to increase false alarm, that is very dangerous so that is not something we want to do. *2010

*That is type 1 error so that is something we do not want to do. *2020

*So you do not want to change power by changing Alpha although that is something under our control. *2023

*Now we could try to change effect size but because of effect size is something that is already sort of true in*2029

*the world right like what we have to do to mess with standard error of the standard deviation of the *2039

*population, we cannot mess with that so this is actually something that is impossible to do. *2045

*So that is one thing that we wish we could do but cannot do anything about. *2052

*Can we change the variability in our sample, can we change the variability? *2067

*Indirectly, we can. *2072

*There is really one way to be able to change standard error. *2075

*Can we do this by changing the standard deviation of the population? *2081

*No, we cannot do that, that is out of our control. *2085

*But we can change N. *2090

*We can collect more data instead of having 40 subjects or cases in our study, we can have 80. *2093

*And in that way we can increase our power and so really the one thing that sort of one tool that sort of *2102

*available to us as researchers in order to affect power is really affecting sample size. *2110

*None of these other things are really that appealing to us. *2116

*We cannot change population variability, we cannot change effect size and if we change Alpha then that is a dangerous option. *2120

*And so what we have left here is affecting sample size. *2133

*Now let us go on to some examples. *2139

*Statistical test is designed with a significance level of .05 sample size of 100. *2144

*As similar test of the same null hypothesis is designed with a significant level of .1 and a sample size of 100. *2149

*If the null hypothesis is false which test has greater power? *2160

*Okay so let us think about this. *2165

*Here we have a situation one test one where Alpha equals .05. *2168

*Test 2 the other test, alpha = .10 so here Alpha is larger. *2178

*Remember alpha is moving that critical test statistic so we have taken this and let us have this Alpha right*2190

*here and what we do is we moved it over, moved it over here, well not that far but just so you can get the idea. *2205

*And now our Alpha is much bigger but what we see is that our beta, our 1 - Beta has also gotten a lot bigger. *2217

*So here we see that power increases but we should also note that now we have a higher tolerance for false *2231

*alarms so we will also have more false alarm, will have more times when we reject the null period so we *2244

*will reject the null lots of time sometimes will be right, sometimes will be wrong, both of this things increase. *2251

*Example 2. *2258

*Suppose the medical researcher was to test the claim of the pharmaceutical company that the mean number of side effects per patient for new drug is 6. *2261

*The researcher is pretty sure the true number of side effects is between 8 and 10 so there like *2270

*pharmaceutical company not telling the whole truth. *2277

*Shows a random sample of patients reporting side effects and chooses the 5% level of significance so Alpha equals .05. *2281

*Is the power of the test larger is the true number of side effects is 8 or 10. *2288

*So let us sort of think about okay what is the question really asking and then explain. *2295

*So is the true number of side effects is 8 or 10 is really talking about your mu? *2302

*And actually, here we are talking about the alternative mu because the null mu is probably going to be six. *2309

*So here is the null hypothesis. *2325

*The null hypothesis is that the pharmaceutical company is telling the truth. *2330

*So the null hypothesis mu is six. *2334

*Now, if the alternative mu is 8, it will be, maybe about here but if the real alternative population is actually*2337

*a 10, so the other alternative, a 10, it is way out here. *2350

*And which of these scenarios is the power larger. *2359

*Well even if we set a very conservative critical test statistic, here is our power for 8 as is the true number of*2365

*side effects but here is the power almost 100% for 10 being the true number of side effects and remember *2382

*I am just trying these with just some standard error I do not care what it is just have to be the same across all of them. *2395

*And so here we see that wow, it is way out farther, more of this is going to be covered when we reject the null. *2402

*And so we see that the power is larger, is the true number of side effects is 10. *2413

*And the reason for that is because this is really a question about effect size. *2420

*The true certain distance between our null hypothesis distribution and our alternative hypothesis distribution. *2428

*We know that as effect size goes up power also goes up easier to detect but we cannot do anything we cannot actually make effect size bigger. *2440

*Example 3. *2455

*Why are both the Z and T statistic affected by N while Colens D and hedges G are not then what do the Z, *2458

*T, D and G all have in common and finally, what commonality does Z and D share. *2469

*What commonality does T and G share? *2478

*Well, I am going to draw this as sort of Ben diagram. *2481

*So let me draw Z here and here, I will drop T and then here I will draw D and it is going to get crazy, here I will draw G. *2484

*Now, if it helps, you might want to think about what these guys mean over the actual population, standard *2511

*deviation over the standard error derived from the population standard deviation. *2523

*And here we have standard error derived from the derived from the estimated population standard *2538

*deviation whereas in D we have the distance, same distance, here just divided by Sigma, here we have the same distance divided by S. *2547

*Okay so why are both the Z and T statistic affected by N while Colens D and Hedges G are not? *2570

*Well, the thing that these two have in common is that these are about standard error and standard error is *2579

*either Sigma divided by square root of N or S divided by square root of N and it is this dividing by square *2587

*root of N that makes these two so affected by N. *2602

*And so it is really because they are distances in terms of standard error. *2607

*So when do the Z, T, D and G all have in common so that is that is the little guy right here, what do they all have in common? *2614

*Well they all have this thing in common. *2627

*So they are all about the distance between sample and population. *2629

*So it is all about that distance. *2641

*Some of them are in terms of standard error and some of them are in terms of population standard deviation. *2644

*So what commonality does Z and D share. *2651

*Well that going to be right in here. *2656

*What do they have in common, they both rely on actually having Sigma. *2658

*T and G both rely only on the sample estimate of the population standard deviation. *2663

*So looks a little messy but hopefully this makes a little more sense. *2671

*Thanks for using educator.com for effect size and power.*2676

*Hi, welcome to educator.com.*0000

*We are going to talk about F distributions today.*0002

*So first we are going to review other distributions recovered besides F, namely the NT.*0004

*Then we are going to introduce the F statistic also called the variance ratio.*0011

*Then we are going to talk about the distribution of all these S, distribution of all these ratios and finally 0024.5 what Alpha and P value mean in an F distribution.*0017

*Because eventually were in a deep hypothesis testing with the F statistic.*0029

*Okay , first, these other distribution so we know how to calculate the Z statistic and we also know how to *0031

*find the probability of such V value in a normal distribution.*0044

*But what is EZ distribution?*0050

*Well, imagine this.*0053

*Take a data set, let us just call it a population.*0056

*We take a data set, I will just draw a circle and we take some sort of sample from it, of size.*0059

*And we actually calculate the Z statistic for this sample so we calculate the goals, get the mean of this little*0068

*sample minus the mu divided by the standard error.*0087

*So you do that and then you plot the Z.*0093

*So imagine you replace all those that sample again to with replacement and you draw another sample and *0098

*you do this again and then you plot that guy and you dump it back in, you draw another sample, you calculate Z.*0115

*So you do that over and over again many times which end up getting is a normal distribution overtime.*0123

*So many times if you plot Z you get a normal distribution and because of that we also call this a Z *0136

*distribution because the distribution made up of a whole bunch of Z and it has the shape of a normal *0150

*distribution so that is what we call a Z distribution.*0159

*Now, if you take that same idea and you do it, you get a sample, and instead of calculating Z for that simply*0162

*you calculate T, if you do this then and then you plot that and you do that over and over and over and over again you get a T distribution.*0175

*And this resulting t-distribution follows the rules of the t-distribution where it depends on the degrees of*0195

*freedom, how wide it is, at the lower your degrees of freedom assertive variable but the higher the bigger *0209

*your degrees of freedom assertive, less variable and more normal it looks.*0217

*And so that is what we call the t-distribution.*0222

*So that is how Z statistic and the Z distribution sort of go together.*0225

*And this is how the T statistic and the t-distribution sort of go together.*0232

*And they just have to imagine taking a whole bunch of this sample, calculating whatever statistic and *0237

*implying that statistic and then looking at the shape of those statistic.*0245

*So really what this is a sampling distribution of Z.*0250

*And this is a sampling distribution of T, instead of using means or Z squares to plot your plane instead use the T statistic.*0260

*And you could do that for anything you could do that for standard deviation and you can do for inter *0281

*quartile, you can make the sampling distribution of anything you want.*0286

*That is important to keep in mind as we go into F distribution.*0290

*Okay so first thing is what is the F statistic?*0293

*We know how to calculate the T statistic and the V statistic, what is the F statistic?*0301

*Well, later on in these lessons were going to come across what we call the ANOVA, the analysis of variance.*0307

*Analyze means to break down and variance is well, you know what variances is, the spread of usually *0316

*around the mean of your data set and so when we analyze variance, we are going to be breaking down *0325

*variances into its multiple component and the F ratio happens to be ratio of those component variances.*0332

*And so I just want you to get sort of the big idea behind the F ratio not exactly how to calculate it, well get*0343

*into the details of that later on but the general concept.*0352

*So the S statistic usually is this idea that we have let us say two samples, x1, x2, x3, y1, y2, y3.*0356

*Now there is always some variation within the sample within exit there is some variation.*0370

*And within the Y there are some variation.*0379

*So there is definitely some variation but there is another variation here that we are really interested in.*0385

*We are really interested in the difference between these two things.*0392

*Between these two samples, so the F statistic really is taking those ideas and turning it into a ratio and here is what a ratio looks like.*0397

*It is really the between sample variance all over the within sample variance.*0408

*I remember variance is always squared, the average squared distance away from the mean and so because *0425

*of that this is a squared number, this is the square number, they are both positive so this number is always going to be greater than zero.*0433

*There is no way that this number could be less than zero so the statistic is always going to be greater than zero.*0442

*Now another way to think about between sample variance and within sample variance is this.*0449

*Whenever we do these kind of test, we are really interested in the differences between the samples like that is really important to us.*0454

*But sometimes their difference is also like a part of that difference is going to be just inherent variation.*0464

*So sometimes there might be a difference between let us say,men and women, or people who got a *0478

*tutorial versus people who did not, right?*0486

*People who study for the test versus people who did not, people went to private school versus people with public school.*0488

*There might be some difference between them?*0495

*But that difference is also going to have variation.*0497

*So this between sample variance often has inherent variation just variance you cannot do anything about *0500

*inherent variation plus real difference the effect size between samples.*0508

*And noticed that we keep using this word between and that is to indicate that part, so between, that is the part that we are really interested in.*0520

*Over within sample variance and so here there is inherent variation between X and between the Y and that *0534

*is not something we are interested in but it is good to know how variable are in the our little samples are.*0557

*Everyone very similar to each other, is very different, we need to compare the difference between the sample to the difference within the samples.*0565

*So this the inherent sample of the within sample variation is just inherent variation.*0574

*So these are all different ways of seeing the same thing and the reason why I want to say I also like this is*0583

*because later on we are not just going to be talking about between sample and within simple differences, we are going to add onto those ideas.*0593

*The final way I want you sort of think about the F statistic is basically this.*0601

*Ultimately in hypothesis testing, where going to want to know about differences between sample, that is the thing that were really interested in.*0608

*So it is going to be the variation that we want to explain because that is the reason that we did our research in the first place.*0616

*All versus the variation we cannot explain, not with this design at least.*0631

*So in our experimental design we will have these two groups and hopefully these groups will be similar to *0646

*each other but different, similar within the group but different between the groups.*0653

*And that is why in a S statistic we want this variation that we want to explain to be quite large and this *0660

*variation that we cannot explain or do anything about to come along for the ride where we want that to be relatively small.*0667

*Okay so let us do a limited thinking about the F ratio.*0676

*Now if we had a very big difference between the groups what kind of F ratio would we have?*0679

*When it is greater than one, less than one?*0688

*Well if our variation between the groups is bigger than the variation within the group then we should have *0690

*a very large F so that should be F that is greater than one right so at least greater than one but maybe a lot 1144.0 greater than 1, it could be 2 over one or 2 over .5.*0697

*Any of those values which show between sample variances are a lot larger than within sample variance.*0708

*And so if there is a lot of within sample variance then that competes with the between sample variance so *0715

*let us say there is a vague between sample difference but there is also a lot of differences within the *0728

*samples themselves and sort of evens out and you might see an F that is smaller or even less than one right if this one is bigger than this one.*0734

*So that is how you could sort of think about the S statistic.*0745

*Now imagine getting that F statistic over and over and over again from the population and plotting a sampling distribution of S statistics.*0748

*What would you get?*0761

*Well, remember that F cannot go below zero because both numbers are going to be positive so the F really stops at zero.*0763

*But this is what the S statistic ends up looking like.*0774

*This is a skewed distribution and it has a positive tail.*0778

*That means it goes for a really long time on the positive side.*0786

*Its one-sided so it is not is not symmetrical, it is actually asymmetrical there is only a positive side and it is*0792

*because of the proportion of variances and variances are positive.*0803

*And like T is a family of distribution and you are going to be able to find the particular F distribution you are*0810

*working with by looking at the degrees of freedom in the numerator, the one about between sample *0819

*differences and by looking at the denominator the sort of leftover or within sample differences variation.*0829

*So you are going to need both of those numbers in order to find out which S statistic you are working with *0847

*and in Excel, it will actually ask you for the degrees of freedom for the numerator and denominator.*0854

*Now let us talk a little bit about what Alpha means here.*0861

*Alpha here, it will still need a cutoff point so critical F instead of a critical T or Z.*0866

*You will still need a critical F and the Alpha will still be our probability of making false alarm given that the null distribution is true.*0877

*This is the null F distribution just saying.*0890

*And the Alpha would be the same thing the probability of false alarm.*0894

*So once you know what that alpha sort of have, how you sort of picture that Alpha, let us talk about what that Alpha actually means.*0899

*If you go back to the original idea for that alpha the original idea is that cut off level.*0910

*So it is our level of tolerance for false alarms.*0924

*How the probability, the false alarm probability that we will tolerate and that is what we want.*0930

*We want Alpha to be very low.*0945

*Now our Alpha will be low, that is the smaller Alpha than this one, our Alpha will be low if our critical F is very big.*0948

*And what does it mean for F to be large?*0962

*This means our between sample variation variability is greater than our within sample variability.*0964

*And that is what it means and so as long as this is much larger than this, we have a large F and that is going*0984

*to mean a smaller a smaller chance of false alarm.*0992

*Now the Alpha is the cutoff level that we are going to set as the significance, the level that we will tolerate.*0998

*So what is the P value?*1007

*So the P value will be given our samples F, this is the probability that we would get this F or higher by chance in this probability.*1009

*So given our samples F actually will be easier so the idea is the probability, the false alarm probability for F*1030

*values, F statistics are equal to or more extreme than our sample, than the F from our sample.*1058

*So the probability that we would get an F greater than the one that we got so F from the sample.*1080

*So this is the F value once we have our sample statistic, this is the probability of false alarm that were willing to tolerate.*1087

*So it is the same idea as T statistics, the alpha, the P value and T statistics, we are just now applying it to a slightly different looking distribution.*1101

*Now examples.*1112

*Why does the F distribution stop at zero but go on in the positive direction until infinity?*1117

*Well, we know why it stops at zero.*1122

*The F distribution is a ratio of two positive numbers and we know that they are positive because variance squared, thus making it always positive.*1125

*But it goes on until infinity because there is no rule that says you can only be this much bigger in the *1148

*numerator than denominator so the numerator can be like infinitely as big as the denominator who could go on forever and ever.*1159

*Example 2, in an F test also called the one-way ANOVA which we are going to talk about in a little bit, the P*1168

*value, you did an F test and the P value is .034, what is the best interpretation of this result?*1177

*It is plausible that all the samples are roughly equal.*1186

*So here we are thinking about let us say two sample and we need this versus this.*1191

*So the F value is between variation over within variation and if we have a big F value, if we have a big *1203

*enough F value, so sample F then we can have a small P value .034.*1229

*So is it possible that all the samples are roughly equal?*1239

*No because we seem to have a large enough between sample variance so I would say no to that one.*1247

*It is possible that all the sample variances are roughly equal.*1256

*Well, that also is not necessarily what this means it could be that these within variations are very similar to*1261

*each other but that is not what this P value is talking about.*1269

*The within sample variation is much larger than the between sample variation.*1272

*Well, it is true we would have a small F instead it is this one.*1278

*The between sample variation is much larger than within.*1283

*So D is our answer.*1286

*Example 3, consider the height of the following pairs of samples.*1288

*Which will have the largest F.*1295

*Which will have the smallest F.*1297

*Okay let us think about this.*1299

*So players from NBA team Lakers versus adults in LA.*1301

*Well, if we draw those two population, Lakers versus LA.*1306

*This probably has a lot of variance, a lot of variance here, that is a lot of people, this probably have a very*1313

*small variance but there is probably pretty sizable difference between those two groups of people right like*1321

*average adult versus like the Lakers were probably all amazingly tall.*1330

*Well so that is the picture here.*1335

*Will this have a larger, will this have a smaller.*1338

*Well, what about adults in San Francisco versus adults in LA.*1341

*Well, this 2 probably both have a lot of within sample variation there's lots of adults in San Francisco, lots of*1348

*adults in a LA, they are all different from each other but their average just should probably be similar, it is*1355

*not like San Francisco's no pursuit for tall people or LA is no pursuit for tall people so this difference *1362

*between the groups will probably be very small but the within group variability will be very large so I would*1368

*guess this would have actually a pretty small F, and what about this one.*1375

*This one is players from an NBA team Lakers versus players from another team and so here we might think *1381

*Lakers, Clippers, and there is probably a pretty small variation here probably everybody is like about 6 feet*1393

*tall, and so they are probably all like super tall so there is not a lot of variation but there also probably similar across the teams to.*1401

*So because probably the average height on the Lakers is probably similar to the average height on the *1416

*Clippers just that they are both tall groups of people so which one of these will probably have the largest F?*1423

*I think the biggest difference between the groups might actually be this one.*1430

*So I would guess I would go at this one given that I am not really sure about the variance here.*1436

*The variance is smaller but I am not sure how to compare these so far.*1447

*So this is the largest F and I am just going to go by having the largest numerator for sure.*1452

*Well, which will have the smallest F?*1460

*As in the smallest F would probably go at this one because not only does it have a small numerator but it *1464

*has extremely large denominator so I would say this one would definitely have the smallest F.*1472

*So that is the end of F distribution.*1478

*See you next time for ANOVAs on educator.com.*1483

*Hi, welcome to educator. com. *0000

*We are going to talk about ANOVA with independent samples today. *0002

*So first we need to talk a little bit about why we need to introduce the ANOVA. *0005

*We had been doing so well at t-test so far. *0011

*Well, there are some limitations of the t-test and that is why we are going to need an ANOVA here. *0013

*An ANOVA is also called the analysis of variance and the analysis of variance is really also could be thought of as the omnibus hypothesis test. *0020

*So still, hypothesis test just like the t-test but it is the omnibus hypothesis test, we are going to talk what that means. *0032

*We are going to need to go over a little bit of notation in order to break down with the ANOVA details. *0041

*And then were really going to get to the nitty-gritty of partitioning or analyzing variance like *0047

*getting down of breaking apart variance into its component parts. *0055

*The we are going to build up the S statistics made up of those bits and pieces of variances and *0059

*then finally talk about how that relates to the F distribution and hypothesis testing. *0066

*Okay so first thing, the limitations of the t-test. *0071

*Well here is a common problem like I want to know this question. *0077

*Who upload more pictures to facebook? *0083

*The Latino users, white users, Asian users or black Facebook users? *0086

*Which of these racial or ethnic groups uploads more pictures to facebook? *0091

*Well, let us see what would happen if we use independent samples t-test? *0098

*What would we have to do? *0101

*Well we have to compare Latinos to white, Latinos to Asian, Latinos to black and whites and Asians and whites and blacks and Asians and blacks. *0104

*As like, all of a sudden we have to do 6 different independent samples t-test. *0111

*That is a lot of tiny, tiny little t-test and really the more t-test you do that increases your likelihood of type 1 error. *0118

*Previously, to calculate type 1 error we looked at one minus the probability that you would be *0127

*correct, so one minus the probability of being right and that was to me like . 05 let say, right? *0135

*But now that we want to calculate the probability of type 1 error for six t-test we have to think *0144

*back to our probability principles but really I just want to look something like this. *0152

*One minus whatever your correct rate is to the sixth power and that is got to be a much higher, *0157

*much higher type 1 error rate than you really want. *0167

*So the problem is that the more t-test you have, the more the bigger the chance of your type 1 *0174

*error and even non-mathematically you could think about this. *0181

*Any time you do a t-test you could reject the null, every time you reject the null you have the *0186

*possibility of making a type 1 error and so if you reject the null six times then you have increased *0193

*your type 1 error rate because your just rejecting more null hypotheses. *0201

*So you should know there are two major limitations of having many many tiny tiny little t-test. *0206

*So you have six separate t-test, one is the increased likelihood of type 1 error and that is bad. *0213

*We do not want a false alarm but there is a second problem, you are not using the full set of data in order to estimate S. *0220

*Remember how before we talked about how estimate of the population standard deviation? *0231

*Well, it would be nice if we had a good estimate of the population standard deviation and you *0237

*know when you have a better estimate of the population standard deviation? *0242

*When you got more data rate when you do a t-test for instance with Latinos than white people *0246

*then you are ignoring your luscious and totally usable data from your Asian and black American *0253

*population so that is a problem you are ignoring some of your data in order to estimate S and *0260

*your estimating S a bunch of different little time instead of having one sort of giant estimate of S *0267

*which would be a better way to go so both of these are major limitations of using many many little t-test. *0274

*So back in the day statisticians knew that there was this problem Ronald Fisher came up with a *0282

*solution and his solution is called an F test for Fisher. *0291

*You think of a new statistic you could name it after you self. *0296

*So he thought of something called an F test but this F test also includes a new way of thinking *0302

*about hypotheses and so the F test could also be thought of as an omnibus test and the way you *0308

*could think about them is like the Lord of the rings ring idea. *0315

*It is one to rule them all instead of doing many many tiny tiny little test, you do one test to *0319

*decide once and for all if there is a difference. *0326

*And because you have this one test you need one null hypothesis and here is what that null hypothesis is. *0329

*You need to test whether all the samples belong to the same population or whether 1 at least *0337

*one belongs to a different population because remember the null hypothesis and the alternative *0346

*hypothesis they have to be like two sides of the same point so your null hypothesis is that they are all equal. *0351

*The mu’s are all equal. *0359

*They all came from exactly the same population. *0360

*The other hypothesis the alternative hypothesis is that they are not all equal but let us think about what that means. *0363

*That means at least two of them are different from each other mean that all of them are 4*0372

*different from each other, that means at least one guy is different from one of these guys. *0377

*That is it ,that is all it means, that is all you can find out. *0382

*So let us consider this situation let us say you have these three samples. *0386

*Your null hypothesis would be that they all came from the same population. *0392

*A1 A2 and A3 all the same population A but if we reject that null hypothesis what have we found out? *0399

*What we found out that at least two of them differ at least, all three of them could differ from each other or it could just be 2. *0413

*It could be that A1 and A2 are the same, the A3 is different. *0421

*It could be the A2 and A3 are the same but A1 is different. *0425

*It could mean that A 1 is totally different from A2 and that is totally different from A3. *0428

*Any of those are possibility so here is a good thing. *0433

*The good thing about the omnibus hypothesis is that you could test all mentioned things at ones. *0436

*That they all come from the same population, you could test that big hypothesis at ones. *0442

*The bad thing about it is that if you reject the null it still did not tell you which populations differ. *0446

*It only tells you that at least one of the valuations is different. *0454

*So when you reject the null, it is not quite as informative but still it is a very useful test. *0459

*So we need to know some notation before we go on. *0466

*An analysis of variance so analysis of variance, that is why it is called the ANOVA so sometimes *0471

*you might do with the opening little ANOVA notation, you want to analyze the variance so when *0479

*we want to analyze the variance we have to really think hard about what variance means. *0486

*And variance as sort of the average spread around some means so how much spread you have. *0492

*Are you really tightly clustered around the mean or you like really just burst around the mean. *0500

*Okay so first things first, consider all the data that we get from all the different groups. *0505

*That is why we have to regroup all the data from all the different groups, and a lot of variance *0511

*around the grand mean and the grand mean is a new idea. *0518

*The grand mean it is not just the mean of your sample but the grant mean is the mean of everybody lock together. *0521

*Pretend there are three groups pretend there is just one giant group that all three data sets have been sort of award into. *0528

*What is the meaning of that giant group? *0536

*That is called the grand mean and so for instance, here is our sample. *0538

*Our sample from A 1 our sample from A2, our sample from A3, and when you have sample means here is what the notation looks like. *0544

*It should be pretty familiar, X bar sub A1, X bar sub A2, X bar sub A3. *0552

*Now when we had a grand mean, we do not have three of them we just have one because remember, they are all lock together, right? *0564

*How do we distinguish the grand mean if we just say X bar we might confuse it for being a *0571

*sample instead of grand mean right and so in order to think grand mean this is the mean of all *0579

*the means, mean of all the samples right, we call it X double bar and that is how we know that it *0585

*is the grand mean so that is definitely one of the things you need to know. *0592

*So now let us talk about partitioning or analyzing the variance. *0596

*When we are analyzing variance, what we want to start with is the total amount of variance. *0606

*First, we got so have the big thing before we jump in apart. *0614

*So what is the big thing, the big variance in the room is total variance and this is the variance *0617

*from every single data point in our giant pool around the grand mean. *0625

*And we can actually just sort of think about how to write this as a formula just by knowing grand *0629

*mean as well as the variance formula right and so variance is always squared distance away from *0639

*the mean divided by however many data points you have to get average square distance from the mean. *0645

*Now we want the distance away from the grand mean so I am going to go ahead and put that *0653

*there instead of X bar I have X double bar and put my data points so that would be Exabyte. *0659

*And we want to get the sum of all of those and then divide by however many data points we have. *0668

*Usually N means the number of data points in a sample. *0676

*How do we tell the N of everybody of all your data points added together? *0682

*Here is how you, you call it N sub total. *0688

*And in this says it is not just the end of one of our little sample because we have three little *0691

*sample, I mean the N of everybody of the total number in your data set. *0698

*And so even this Exabyte I do not really mean just the axis in sample 1, I mean every single data *0704

*point so I would say I goes from 1 all the way up to N total. *0713

*Sorry this is a little small, N’s of total appear and so this will cycle through every single X, every *0719

*single data point in your entire sample lumped together. *0729

*Get their distance away from the grand mean, square it, add those squared distances together *0732

*divide by N so this is just the general idea of variance. *0741

*Average where distance from the mean. *0747

*In this case, retirement grand mean and so how do we say total variance? *0750

*Well it would be nice if we could say like, oh this is something subtotal, right? *0757

*Before we go on to variance though I just want to stop here before we go into average variance, *0765

*I just want to talk about this thing, what is this thing? *0773

*And so let us talk about some of variance request so variance is always going to be the sum of *0777

*squared distances, sum of squares divided by N or if you are talking about S, S squared is the sum *0784

*of squared distances over N minus 1 and another way of saying that is SS over degrees of freedom. *0795

*So we are just going to stop here for a second and just talk about this sum of squares and we are going to call that sum of squares total. *0805

*So that sum of squares total and that is going to be important to us because later we are going to *0817

*used these sum of squares, these different sum of squares to then talk about variance. *0824

*It is the squares are very unrelated to the idea of variance. *0830

*Now we have this total variance because this is really the idea of how much you are varying. *0834

*We have this total variance and were going to partition it into two types of variance. *0840

*One is within group variation and the other is between group variations. *0845

*So we have 3 groups, the between group variance is going to look at how different they are from each other. *0850

*The with in group variance is just going to look at how different they are from their own group, *0860

*how different the data are from their own group and that is going to be important because this *0865

*sum of squares total actually is made of sum of squares within plus sum of squares between. *0871

*So because of this idea we can really now see, where taking us all variance and partitioning it *0882

*into within group variance between group variance or between sample variance. *0892

*So first things first, within group variance. *0899

*How do we get an idea of how different each sample is from itself. *0902

*Well the very idea is just like what we have been talking about before. *0912

*This is each samples variance around their own mean and we already know the notation for this mean. *0917

*So that would be something like how much does everybody in sample A1 differ from the mean of *0938

*A 1 so what is that different to getting the sum of squares. *0947

*And what is the variance, the sum of squares for everybody in A2 square and the same thing for everybody in A3. *0954

*So this is sort of the regular use of variance that we used before regular use of sum of squares that we have used before. *0971

*Just looking at each sample variance from its own sample mean. *0977

*Now how do we get between group variance? *0982

* Between group variance is going to be each samples mean, how much does it very from the *0986

*grand mean, difference, squared difference from grand mean so there is some grand mean and *0999

*how much does each sample mean differ from that grand mean. *1011

*And so that is going to be between group variation. *1016

*How much do the group differ from that grand mean. *1020

*So first of all let us just review variance and sum of squares. *1024

*So sum of squares is the idea that were in use over and over again and it is just this idea that *1033

*yours summing the sigma sign, sum from X bar squared. *1043

*So it is just basically that the squared distance away from the mean and add them up. *1052

*That is sum of squares. *1061

*Now what we are doing is we are sort of swapping out this idea of the mean for things like grand *1063

*mean, sample mean, and were also swapping out what our data points are. *1071

*Is this like from N total, is it from all of data points, is it just the end from the sample, is it the group means? *1082

*So were swapping out these two ideas in order to get our sum of squares total, sum of square *1098

*between or sum of squares within but it is always the same idea. *1106

*Sum of distance, squared, add them up. *1110

*Okay now so what is variance in relationship? *1113

*Well variance is the average squared distance and so in doing this we always take the sum of *1116

*squares and we divide by however number we own but how many data points we have? *1130

*But often where using estimates of S instead of actually having the population standard deviation. *1139

*So were going to be using degrees of freedom instead of just N and we have different kinds of *1146

*degrees of freedom for between and within group variation so watch out for them. *1153

*Okay now let us go back to the idea of the F statistic. *1162

*Now that we have broken it down a little bit in terms of what kind of different variances there *1167

*are, hopefully the F statistic makes a little more steps sense. *1171

*The idea is that you want to take the ratio of the between group or sample variance over the *1175

*within group variance and the reason we want this particular ratio is that were actually very *1187

*interested in the between group difference that what our hypothesis test is all about whether the groups or difference are the same. *1197

*The within group variation, we cannot account for. *1206

*Its variation that just is inherent in the system and so we need to compare the between group *1210

*variation which we care about with the within group variation we cannot explain we do not have *1218

*any explanation for at least not in this hypothesis test, we have to do other tests to figure out that. *1223

*Okay so now that were here we need to do is replace these conceptual ideas with some of the things that we have been learning about. *1230

*In particular the variance between the variance within and so variance we are going to use S squared but S squared between over S squared within. *1242

*So variance between over variance within but now we know a little bit like we have refreshed, *1260

*what is variance about, how can we break it down in terms of sum of squares? *1266

*Well, that is what we are going to do. *1272

*We are going to double-click on this guy and here is what we see inside. *1276

*We see the sum of squares between divided by the degrees of freedom between all over the *1280

*sum of squares within then divided by the degrees of freedom within and this is how were going to actually calculate our S statistic. *1291

*Now, we will write out the formulas for each these but it is sort of good to know like where the S *1301

*statistics are comes from its conceptual route, you always wanted to be able to go back there. *1309

*Because ultimately when we have a large F, we want to be able to say, this means there is a *1314

*larger between group variation then within group relative to within group variation. *1321

*A larger difference in the thing that were interested in over the variance that we have no explanation for. *1327

*Okay so now let us figure out how to break down this idea and remember this idea really is the breakdown of the variance between. *1332

*So were breaking down the broken down thing. *1343

*So conceptually what is this? *1347

*Well, conceptually this is the difference of sample mean from the grand mean so imagine our *1350

*little group and their sum grand mean that all of these guys contributed to but this all have a little sample mean of their own. *1357

*What I want to do is know the difference between these, squared, then add them up. *1376

*That is the idea behind this. *1384

*So first of all how many means do we have how many data sets do we have, how many data points do we have? *1386

*Well we have a data point for every sample that we have so how many means do we have? *1395

*Or how many samples do we have. *1402

*We actually have a term for that. *1404

*The above letter that we reserve for how many samples is K, number of samples. *1406

*And so that you could think about okay if that is the number of samples then what might be the degrees of freedom here? *1415

*Well, just going to be K -1, here is why. *1427

*In order to get the grand mean we could do weighted average of these means and since there *1434

*are three of them if we knew what two of them were in advance the third one would not be free *1442

*to vary, we lockdown with that third one. *1449

*So the degree of freedom is K – 1. *1451

*Okay so what is the actual sum of squares between and know you need to take into *1454

*consideration how many actual data points are in each group. *1463

*For instance, group one might have a lot of data point or two might only have a few data points which means should matter more. *1468

*Well that can be taken into account. *1476

*So first things first, how do we tell it get the difference between this mean and this mean. *1479

*That is going to be this. *1486

*X bar minus X double bar so get them the difference between the mean and the grand mean. *1489

*Now we several means here so I am going to put an I for index and in my sum of squares my I is *1497

*going to go from one up through K so for each group that I have I want you to get this distance and square it. *1507

*Not only I am going to stop there but I also want you to make it count a lot if it has a lot of data *1515

*points so if this guy have a lot of data point he should get more votes, his difference from the *1526

*grand mean should count more than this guys different and so that is what we get by multiplying *1531

*by N if N is very large, this distance is an account a lot if N is very small, this distance is not going to count as much. *1538

*And this is the sum of squares between so that is the idea. *1546

*Okay so now we actually know this and this so we could actually create this guy but putting these two together. *1554

*Now let us talk about sum of squares within now that we know sum of squares between pretty well. *1563

*Well, first thing we need to know is that this idea sum of squares within divided by degrees of *1582

*freedom within is actually going to give us the variance within. *1587

*Let us talk about what this means conceptually. *1593

*This means the spread of all the data points from their own sample mean. *1596

*So this is the picture I want you to think of. *1604

*So everybody has their own little sample mean, X bars, own little sample mean and here are my *1610

*little data point and I want to get the distance of each set away from their own sets mean. *1620

*This is going to give me the within group variation. *1629

*Well, we need to think about first, how many data points do we have? *1635

*Well we have a total of N total, because you need to count all of these data points you need to add them all up. *1643

*The total number of data point. *1652

*So what is the degrees of freedom? *1656

*Well, it is not just N total -1. *1659

*How many means did we find? *1661

*We found three means, for each time we calculate a mean, we loss a degrees of freedom so it is *1663

*really the N total minus the number of means that we calculate and here, it is 3, it is 3 because we have three groups. *1674

*Remember, we have a letter for how many groups we have, and that is K so it is really going to *1684

*be N total minus K the number of group and that is going to give us the degrees of freedom within. *1689

*So what is the sum of squares within? *1698

*The sum of squares within is really going to be the sum of squares here plus the sum of squares here plus last the sum of squares here. *1701

*So for each group just get the sum of squares. *1713

*That is a pretty easy idea so the sum of squares within is just add up all the sum of squares. *1718

*Now what it this I mean? *1728

*I means the sum of squares for each group and that is I going from one to K so for however many *1730

*groups you have get that group sum of squares added to the next group sum of squares added to *1740

*the next group sum of squares and these are general formulas that work for two groups three *1746

*groups four groups, so that is sum of squares within and now that we know this and this, we could calculate this. *1751

*So now let us put it all together all at once. *1764

*My apologies because this may look a little bit tiny on your screen but hopefully you could sort of *1770

*reconstruct it from when you seen before because I am writing the same formulas just in a *1781

*different format just to show you how they all relate to each other. *1786

*So first conceptually this is always important because you can forget the formula but do not *1789

*forget the concept because from the concept you could reconstruct the formula. *1796

*It does take a little bit of mental work that you can do. *1800

*So first things first, the whole idea of the F is the between group variation over the within group variation. *1803

*So that is the whole idea right there and in order to get that we are going to get the variation between over the variability within. *1817

*Actually, I wrote this in the wrong place, should have written it down in the formula section. *1831

*So F equals the variability between divided by the variability within. *1839

*So that is the F. *1852

*Now for the F you cannot just calculate the sum of squares because really, the F is made up of a *1856

*bunch of squares and for F you actually need 2 degrees of freedom and that is going to be *1861

*determined by the between group degrees of freedom in the within group degrees of freedom. *1865

*So these I am just going to leave them empty. *1871

*Now let us talk about between group variability. *1873

*The big idea of this is that this spread around of sample means, around. *1876

*So gonna put S there of ex-bars around the grand mean. *1891

*That is what we are really looking for, that idea of this spread of all the sample means around the grand mean. *1897

*However the within group variability is the spread of data points from own sample mean. *1904

*So for each little group, what is the spread there? *1920

*So that is the idea of these two things. *1923

*Now in order to break it down into the formula you first wanted to get into what is S squared *1928

*between, so if you double-click on that, that takes you here, you double-click on this one, it will take you here S squared within. *1935

*So the variance between the between group variability, this is going to be just the very basic idea of variance. *1943

*Sum of squares over degrees of freedom. *1955

*Same thing here, sum of squares over degrees of freedom. *1958

*That stuff you already know but the only difference is with a little between here and with a little within here so that is only difference. *1963

*Once you get there then you could break this down right and you could say sum of squares *1973

*between and if you forget what the formula is, you can look up here, spread of ex-bars around the grand mean. *1978

*So X bar minus grand mean. *1988

*You know you have a whole bunch of them, sum of squares and you are going to go from 1 up *1990

*through K that is how many sample means you have. *2000

*And you wanted to be waited. *2005

*You wanted to count more your distance counts more if you have more data points in your *2008

*sample and then the degrees of freedom is fairly straightforward. *2016

*It is the number of the means -1 because when you find out your grand mean it is going to limit *2021

*one of those guys so your degrees of freedom is lessened by one. *2030

*So for sum of squares within, let us go back to this idea that spread of all the data point away *2034

*from their own sample mean and that is just going to be all those sum of squares for each little *2043

*group and you already know from for that, added together. *2051

*So I goes from one up to K. *2055

*And the degrees of freedom is really just this idea that you have all these points, all this data *2058

*points and total minus however many means you found because that is going to limit the *2072

*degrees of freedom for those data point and that is K. *2080

*One another thing I want to just say right here, it is just this idea that you might see in your *2083

*textbook or in a statistics package this idea called mean squared error. *2091

*So this term right here is sometimes going to be called the mean squared error term so that a common thing that you might see. *2099

*This may be called mean squared between or you might just see the mean square between *2112

*groups or something like that so between group start might be written out. *2126

*But almost always this denominator is going to be called mean squared error. *2130

*The reason I want to mention it here is not only to connect this lesson with whatever is going on *2135

*on your classes but also because mean squared error will be an important term for later on when *2142

*we are going to other kinds of ANOVA. *2148

*So now let us get to examples. *2151

*So first who uploads more photos? *2156

*People of unknown ethnicity Latino Asian black or white Facebook users. *2158

*So what are null hypothesis and sorry, you might be like, how will I ever know? *2164

*Is this data set found in your downloads? *2172

*And so the download looks like this and there is however many uploaded photos so here is *2176

*uploaded photos here so this person has uploaded 892 photos and their ethnicity is at zero. *2185

*And zero is just a stand-in for the term unknown or blank so they may have left there blank. *2191

*So the Latino sample is one, the Asian sample is 2, the black or African-American is three, the whiter European-American sample is 4. *2198

*And so you can look through that data set, I kind of recorded that just so that we can easily through see where we are. *2210

*Okay let us start off with our hypotheses. *2217

*On this hypotheses the hypotheses to rule them all right the null hypotheses should say that all *2222

*of these ethnicities and even unknown are all the same when it comes to uploading photos. *2231

*So our mu of ethnicity zero, occultist zero, occultist 1,2,3,4 only because that is what is also in the data set. *2239

*The mu of ethnicity zero, the mu of ethnicity 1 equals the mu of ethnicity 2, the mu of ethnicity three, equals the mu of ethnicity 4. *2251

*So we could say this in order to, say look they are all the same mathematically. *2265

*So this is how you write out that idea of they are all the same, they all came from the same population. *2276

*The reason we want to use E0 E1 E2 is just that it is going to make it a lot easier for us to write *2281

*the alternative hypotheses and this also helps us keep in mind why are we comparing the different groups. *2291

*What is the variable they will differ on and the variable is ethnicity and they all differ on that *2298

*variable they will have different values of it and that the between subjects variable so at least in *2307

*our sample people are either Latino or Asian or black or white although they can be both, just not in our sample. *2315

*So the alternative hypotheses is that the mu sub E are not all the same, not all equal. *2323

*We do not actually put does not equal because we know whether it is easy to that are not equal *2346

*or these two that are not equal or this one and this one that is not equal right. *2363

*So we do not make those claims and that is why you do not want it right those not equal ones *2367

*you want to just write a sentence that the means are just not all the same. *2371

*Now at the site in a significance level, just like before let us decide on a significance level of . 05, it is commonly accepted. *2376

*And because we are going to be calculating an S statistic, were going to be comparing it to disc alpha. *2384

*So it is always one tail, always only on the positive tail and so this is the F distribution. *2397

*Okay now let us talk about the decision stage so in the decision stage you want to draw the F distribution, just like I did so here is alpha, here is zero. *2404

*We need to find the critical F but in order to find the critical F we actually need to know the two *2419

*different degrees of freedom because this distribution is going to be different based on those 2° of freedom. *2434

*So we need to know the degrees of freedom in the numerator which in this case is the degrees of *2441

*freedom between and the degrees of freedom in the denominator and that is going to be the *2448

*degrees of freedom within, we could actually calculate that. *2457

*The degrees of freedom between is K -1 and here our K is 12345, K equals 5, 5 groups so that can *2460

*be a degrees of freedom of 4, and the degrees of freedom within is going to be N total minus K. *2473

*And so let us see how many we have total. *2484

*So we could do, we could just do account if you go down here I have actually sort of filled it in for *2488

*you a little bit just so that it is nice and formatted, I used E1 2345 but that really mean, one of them should be easy zero. *2500

*So K is five, we have five different groups, the degrees of freedom between is going to be 5-1, *2511

*the degrees of freedom within, we are going to need to know the total number of data points we *2520

*have so we need to count all the data point that we have. *2527

*All these different data point minus K so here is K. *2531

*So that is a 94 so apparently we have 99 people in our sample. *2541

*So then we can find the critical F. *2547

*Now ones we have the degrees of freedom between and the degrees of freedom within here just *2550

*to remind you this is the numerator and this is the denominator degrees of freedom. *2555

*Once we have that you can look it up in the back of your book. *2561

*Look for the F distribution chart or table and you need to find one of the, either the columns and *2564

*rows usually the columns will say degrees of freedom numerator and the degrees of freedom *2574

*denominator and then you could use both to look up your critical F 5% or you can look it up in Excel. *2580

*And the way we do that is by using F in because F discs will give you the probability, F in you put in probability and get the F value. *2594

*So the probability is . 05, only one tail so we do not have to worry about that. *2607

*The first degrees of freedom we are looking for is the numerator one and the second degrees of *2611

*freedom we are looking for is the denominator one. *2615

*And so when we look at that we see 2. 47 to be a critical F. *2620

*So your critical F is 2. 47 and so we need an F value greater than that or a P value less than . 05 in *2633

*order to reject our null hypothesis that they are all the same, all come from the same population. *2644

*Okay so step 4 in our same question. *2650

*We need to calculate the sample statistic as well as the P value so in order to calculate the *2658

*sample statistic we need to calculate F because F is the only test statistic that will help us rollout our omnivorous hypothesis. *2666

*Remember that is going to be the variance between over the variance within. *2675

*And once we get our F, then we can find the P value at that F. *2681

*So what is the probability of getting an F value that big or bigger given that the null hypothesis is true. *2688

*And we want that P value to be very small. *2697

*So let us go ahead and go to our example. *2700

*Example 1 and here I have already put in these formulas for you but one thing that I like to do for *2706

*myself is I like to tell myself sort of what I need and so I need this and then I break it down one *2715

*row at a time, the next row is going to be assessed between over the degrees of freedom *2722

*between and then I can find each one of those things separately and then I also am going to *2730

*break down the variance within into the sum of squares within and degrees of freedom within and I break those down. *2736

*Okay so first things first, I want to find the variance between but in order to do that I need to find *2743

*sum of squares between and that is this idea that I get every mean, so I need the mean for every *2750

*single one of these groups, for the mean for unknown, mean for Latino users for Asian users and *2758

*so on and so forth and I need to find the grand mean. *2764

*I need to find the squared distances between those guys. *2768

*Okay so first, I need to know how many people are in the this particular sample. *2770

*So let us find the count of E0. *2781

*That is our zero ethnicity for unknown people. *2785

*So I am going to account those people, and then I also going to count E1 and also going to count *2791

*E2, I am also going to count my E3 and finally I am going to count my E4. *2807

*Now these are the same data point that I am going to be using over and over again so what I am *2830

*going to do is I am going to lockdown my data point. *2845

*Say use this data whenever you are talking about the E subzero. *2848

*Use this data whenever I am talking about E1. *2854

*Use this data whenever I talk about E2 and use this data whenever I talk about E3, use this data when I talk about E4. *2862

*Now the nice thing about this is that you could see that they almost all have 20 data points in each sample. *2879

*The only one that differs is the unknown population the unknown ethnicity sample and they are just off by one. *2891

*So, what is the meaning of the sample? *2900

*One thing I could do is I could just copy and paste the cross but what I really want to do is I do *2904

*not want to get the count anymore, I want to get the average. *2915

*So once I do that I could just type an average instead of count, save me a little bit of work and I find all these X bars, X bars for 01234. *2918

*Now let us find the grand mean. *2941

*The grand mean it is going to be the means for everybody so that is going to be the average for every single data point that we have. *2944

*And we really only need to find the grand mean ones. *2951

*If you want you could just point to the grand mean, copy and paste that down it should be the *2962

*same grand mean over and over again or you could just refer to this top one every single time. *2972

*So now let us put together our N times the distance squared before we add them all up. *2978

*So we have N times the distance X bar minus the grand mean, square that, and that is a huge *2990

*number, variance and now we are going to sum them all up. *3004

*Equal sign, sum and I want to sum all of this up. *3011

*I get this giant number 8 million something. *3019

*So huge number. *3023

*So once I have that I can just put a pointer here. *3025

*I just put equal sign and point to this sum. *3031

*And that is really the sum of squares between. *3035

*What about degrees of freedom between, have I already found that? *3039

*Yes I have, I found it up here. *3047

*So I am not going to calculate that again I am just going to point to it. *3049

*Once I have these two now I can get variance between groups. *3054

*So it is this divided by the sum of squares divided the degrees of freedom. *3060

*We saw the giant number that it make sense if you take 8 millions something divide by 4 you get 2 millions something. *3067

*It is still a giant number but is it more giant than the variance within? *3072

*I do not know, let us see. *3080

*So in order to find the variance within then I need to find the sum of squares within as well as the degrees of freedom within. *3082

*So how do I find sum of squares within? *3087

*Well, one thing I could do I could go to each data point and find mean, subtract each X from each *3093

*mean, square it, add them all up, or I could use a little trick. *3101

*I might use a little trick. *3107

*So just to remind you. *3113

*So here is my little trick. *3113

*So remember the variance of anything, the variance is going to be some of squares divided by N-1. *3116

*So if I find variance and I multiply it by N-1 I could get my sum of squares, I could do variance times N-1. *3129

*I could use that trick if I use XL. *3146

*So here is what I am going to do. *3152

*I am going to find the variance. *3157

*First it might be easy if I copied these. *3159

*Just so that I do not have to go and select those. *3166

*If I find the variance and then I multiply it by N -1, I get my sum of squares. *3171

*I am just working backwards from what I know about variance. *3186

*So I am going to do that same thing here and and get my variance and multiply it by N minus 1. *3189

*Get my variance multiplied by N – 1, finally variance multiplied by N – 1. *3199

*Obviously you do not have to do this, you could go ahead and actually compute sum of squares *3234

*for each set of data but that would take up a lot of room and typically more time so if XL is *3243

*handy to you then I really highly recommend the shortcut and then we will just want to sum all the guys up. *3251

*That some of all the sum of squares and we get this giant number. *3258

*We get 42 million, really large number. *3263

*But our degrees of freedom within is also a larger number than our degrees of freedom between. *3279

*And so if I find out my variance within then let us see. *3287

*Is this smaller or bigger. *3295

*Well we see that this number 450,000, that is the smaller number than 2 million so that is looking good for S statistic. *3297

*So our S statistic is the variance between divided by the variance within and we get 4. 48 and *3312

*that is quite a bit larger than our critical F of 2.46 and I have forgotten to put a place for P value *3323

*but let us calculate the P value here so in order to calculate P value we put F discs and we put in *3334

*the F value and the degrees of freedom for the numerator as well as the degrees of freedom for the denominator. *3343

*And we get P = .002 just quite a bit smaller than .05. *3353

*So that is a good thing so in step five we reject the null. *3362

*How does which group is different or multiple groups are different from each other? *3366

*We just know that the groups are not all the same that is all we know. *3374

*Okay so we got a P value equals .002 so we rejected the null hypothesis. *3378

*Here is the thing, remember at the end of this, we still do not know who is who, we just know that somebody is different. *3390

*At the end of this what you wanted to do is, there is going to be like little paired t-test. *3398

*They are often called contrast and you want to do that in order to figure out what your actual, *3405

*which group actually differs from which other group not just whether some group differs from *3414

*some other group and so you want to do a little bit more after you do this. *3420

*This are called post hoc test. *3425

*And in a lot of ways they are very similar to t-test were you look at pairs. *3427

*There is one change, they change the sort of P value that you are looking for so but you wanted *3439

*to do the post hoc tests afterwards and to do all the little comparison so that you can figure out who is different from who. *3446

*But you are only allowed to do a post hoc test if you rejected the null hypothesis. *3452

*So you are not allowed to do a post hoc test if you have not reject the null hypothesis that is why *3457

*we cannot just get to the step from the very beginning. *3464

*So first thing we need to do if you reject is do post hoc test. *3468

*Something you need to do is find the effect size. *3472

*In the case of an F test, you are not going to find like coens D or hedges G. *3475

*You are not going to find that kind of effect size. *3486

*You are going to find what it’s called Eta squared. *3488

* Eta squared, it looks like the N squared. *3490

*And eight is where it is going to give you an idea of the effect size. *3495

*Now let us go to example 2. *3499

*So also the data is provided in your download. *3504

*A pharmaceutical company wants to know whether new drug had the side effect of causing patients to become jittery. *3508

*3 randomly selected sample, the patients were given 3 mild doses of the drug. *3513

*0, 100 200 mg and they were also given a finger tapping exercise. *3518

*Does this drug affect this finger tapping behaviour? *3523

*Will this one I did not format really nicely for you because I want to serve figured out as you go but do not worry I will do this with you. *3527

*So first things first Omnibus hypothesis. *3536

*And that is that all three dosages are the same so mu of dosage zero = mu of dosage 100 = mu of dosage 200. *3538

*And the alternative hypothesis is that mu of the dosages are not all same. *3563

*Okay step 2. *3575

*Alpha = .05. *3579

*Step three decision stage how do we make our decision to reject or fail to reject. *3581

*First you want to draw that F distribution, put colour in that Alpha = .05, that is the error rate were willing to tolerate. *3591

*Now what is our critical F? *3603

*In order to find our critical F we need to know the degrees of freedom for between the degrees of freedom for within. *3607

*So if you go to example 2 the worksheet for example 2, example 2 then you can see this data set. *3615

*Now usually this is not the way data is set up that especially if you use SPSS or some of these other statistics packages. *3627

*Usually you will see the data for one person on one line just like this. *3635

*Just like example 1 the data for one person their ethnicity and their photos are on one line. *3641

*You will rarely see this but you may see this in textbooks so I do want you to sort of pay attention *3649

*to that but here and the problem is that different people were given the different dosages so you *3655

*could assume each square to be a different person. *3660

*So, were on step three decision stage and in order to figure out our critical F, we need to know *3663

*the degrees of freedom between and degrees of freedom within, that is not so pretty anymore, *3677

*this takes a long time to do that to put all the little fancy things in there but it is very easy. *3683

*So degrees of freedom between in order to find that it would be really helpful if we knew the K, *3695

*how many groups right and there are three groups, three different drug dosages. *3699

*So it is K -1 degree of freedom 2. *3705

*In order to find degrees of freedom within we need to know N total. *3710

*How many total data points do we have? *3716

* And we could easily find that in XL using count and selecting all our data point so there is 30 people 10 people in each group. *3719

*So that is going to be N total minus K. *3730

*That should be 27. *3736

*Once we know that we can find our critical F and use F in probability of .05 degrees of freedom *3738

*for the numerator is going to be degrees of freedom between, degrees of freedom for the *3747

*denominator is degrees of freedom within and we get 3. 35 as our critical F. *3752

*Note that this is a larger critical F than before when we had more data points. *3760

*Like 90 data points in the other example and because of that brought down our critical F. *3767

*Now let us go to step 4, step 4 we need to calculate the sample F as well as the P value. *3772

*Let us talk about how you do F. *3784

*Here we need the variance, variance between divided by the variance within. *3786

*How do we find the variance between? *3795

*Well that is going to be the sum of squares between divided by the degrees of freedom between. *3797

*How do we find sum of squares between? *3805

*Well remember, the idea of it is going to be the means for each group, distance from that mean *3809

*to the grand mean, square that distance, weight that distance by how many N we have, and then add them all up. *3816

*So in order to get that, up and down here what will we put in the other stuff to? *3826

*The variance within, that is going to be the sum of squares within divided by the degrees of *3837

*freedom within, just so I know how much room I have to work with. *3844

*Okay so first they might be helpful to know which being were talking about, the dosage so it is 0, *3849

*D0, D 100 and D200, those are three different groups. *3857

*What is the N for each of these groups, what is the X-bar for each of these groups, what is the *3865

*grand mean and then we want to look at N times X bar minus the grand mean, we want to *3872

*square that and then once we have that, now we want to add these up and so I will put sum here *3883

*just so that I can remember to add them up. *3894

*Okay so the N for all of these are 10, we already know that, and let us find the X-bar. *3897

*So this is the average of this and then the next one, it is the same thing, we know it is the same *3906

*thing the average except for column B and the next one is average again, for column C for 200. *3922

*How do we find the grand mean? *3934

*We find the average, we could put a little pointer so that they all have the same grand means. *3937

*Now we could calculate the weighted distance squared for each of these group means. *3952

*So it is N times X bar minus the grand mean, squared. *3962

*And once you have that you could just dragging all the way down and here we sum of these all up. *3970

*We sum these weighted differences up and we get a sum of squares of 394. *3983

*And we already know that degrees of freedom between group so we could put this in divided by this number. *3994

*We get 197. *4006

*Now let us see. *4009

*It is not going to be bigger or smaller than the variance within right and in order to find the *4011

*variance within it helps to just sort of conceptually remember, okay what is the sum of squares *4018

*within, then the sum of squares for each of these groups from their own means. *4022

*And so the sum of squares for each of these dosages are going to be, and I am just going to use *4029

*that shortcut, the variance for this set multiplied by nine, that is N -1. *4041

*And I am just going to take that and I am going to say do that for the second column as well as the third column. *4058

*And once they do that I just want to sum these all up and I get 419. *4073

*So now I have my sum of squares within. *4082

*I divide that by my degrees of freedom within and I get 15.53 and even before we do anything *4085

*we could see wow that variance between is a lot bigger than the variance within. *4095

*So we divide and we get 12.69, 12.7 right and that is much larger than the critical F that we set. *4099

*What is the P value for this? *4111

*We use F disc, we put in the F value we got, we put in the degrees of freedom between, degrees of freedom within and we get a P value of .0001. *4114

*So were pretty sure that there is a difference between these 3 groups in terms of finger tapping. *4131

*We just do not know what that difference is. *4138

*So step five would be reject null and once we decided to reject the null then you would go on to *4140

*do post talk test as well as calculating effect size. *4153

*So that is one way ANOVA with independent samples. *4157

*Thanks for using educator.com. *4163

*Hi, welcome to educator.com.*0000

*Today we are going to talk about repeated measures ANOVA.*0002

*So the repeated measures ANOVA is a lot like the regular one way independent samples ANOVA that we have been talking about.*0004

*But it is also a lot like the paired samples t-test and so we are going to talk about why we need the repeated measures ANOVA .*0014

*And we are going to contrast the independent samples ANOVA with the repeated measures *0022

*ANOVA and finally we are going to breakdown that repeated measures at statistic into its component variant parts.*0027

*Okay so previously, when we talked about one-way ANOVA we talk initially about why we *0035

*needed it and the reason why we need ANOVA is that the t-test is limited.*0044

*So previously we talked about this example, who uploads more pictures, Latino white Asian or black Facebook users? *0049

*When we saw this problem and we thought about maybe doing independent samples t-test we realize we would have to do a whole bunch of little t-test.*0058

*Well let us get this problem.*0066

*It is similar in some ways but it is also a little bit different so here is the question.*0070

*Which prototype is most frequently used on facebook? *0076

*Tagged, uploaded mobile uploads for profile pictures? *0079

*Now in the same way that this has many groups, at this all the problem also has many groups, *0083

*the one thing you could serve immediately tell us of that is if we try to use t-test we also have to use a bunch of little t-test here.*0091

*But here is another thing.*0099

*These variables are actually linked to one another.*0102

*Often people who have tagged photos have a number of uploaded photos who have a number of *0104

*mobile uploads will also have a number of profile pictures.*0110

*So in this sense although these are made up of four just separate groups of users and the user *0114

*here is the linked to any of the users and the other groups Latino, white, Asian, black groups, here we have these four sets of data.*0121

*Tagged, uploaded mobile or profile pictures but the number of tagged photos is linked to some *0135

*number of uploaded photos probably because they come from the same person and maybe this *0146

*person owns the digital camera that they really loving carry around everywhere.*0153

*So these scores in these different groups are actually linked to each other and these are what we *0158

*have called previously dependent samples or we called them paired samples before because *0167

*there were only two groups of them at that time but now we have four groups but we could still see that linked principle still hold.*0173

*So here were talking about were talking about different samples, multiple numbers samples *0181

*more than two but these samples are also linked to each other in some way.*0188

*And because of that those are called repeated measures because we are repeatedly measuring something over and over again.*0194

*Measuring photos here measuring photos here measuring photos here measuring photos here and because that is called repeated measures.*0204

*It is very similar to the idea of paired samples except were now talking about more than two.*0211

*So 3, 4, 5 we call those repeated measures so we have the same problem here as we did here.*0217

*If we have a bunch of t-test of our solution is a bunch of t-test, we have two problems whether their paired t-test or independent samples.*0225

*So in this case they would be paired.*0243

*That even in the case of paired t-test of the same problems that we did before, the first problem *0247

*is that with so many t-test the probability of false alarms goes up.*0254

*So this is going to be a problem.*0258

*And it is because we reject more null hypotheses every time you reject a null hypotheses you have a .05 chance of error.*0264

*So where compounding that problem.*0273

*The 2nd thing that is wrong when we do a whole bunch of little t-test instead of one giant test is *0277

*that we are ignoring some of the data when where calculating the population standard deviation.*0284

*So what we estimate that population standard deviation the more data of the better but if we *0291

*only look at two of the sample that a time then were ignoring the other two perfectly good sets *0297

*of data and were not using them in order to help us estimate more accurately the population standard deviation.*0303

*So we get a poorer estimate of S because we are not using all the data at our disposal.*0311

*So that is the problem and we need to find a way around it, thankfully, Ronald Fisher comes to the rescue with his test.*0338

*Okay so the ANOVA that is our general solution to the problem of too many tiny t-test. *0349

*But so far we only talked about ANOVAs for independent samples.*0356

*Now we need an ANOVA for repeated measures so the ANOVA is always going to start the same way with the Omnibus hypothesis.*0360

*One hypothesis to rule them all and the Omnibus hypothesis almost said all the samples come from the same population.*0370

*So the first group of photos equals the mu of the second group of photos equals the mu of the *0378

*third group of photos equals the mu of the fourth group.*0389

*And the alternative hypothesis is not that they are not all not equal to each other but that at least one is different, outlier.*0392

*And so the way we say that is that all mu’s of P, all the mu’s of the different photo type are not the same.*0404

*Now we have to keep in mind the logic that all of these mu’s are not the same, is not the same as *0421

*saying all of the mu’s are different from each other.*0430

*And when we say all of them are not the same if even one of them is not the same then this alternative hypothesis is true.*0433

*So this starts off much the same way as independent samples from there we go on to analyze variance.*0442

*And here were going to use that S statistic again.*0450

*And Ronald Fisher's big idea that he had upon is this idea that it when we talk about the F it is a *0458

*ratio of variances and really one way of thinking about it is the ratio of between sample or group variability over the within sample variability.*0469

*And another way of thinking about this is if the variability we are interested in and I do not just *0495

*mean that over passionate about it or we find a very curious but I really need is the variability *0510

*that we are making a hypothesis about over the variability that we cannot explain.*0515

*We do not know where that vary the other variability comes from, it just exists and we have to deal with it.*0523

*Okay and so this S statistic is going to be the same concept, the same concept will going to come *0536

*up, again we will talk about the repeated measures version of F.*0544

*There are going to be some subtle differences though.*0548

*Okay so let us talk about the independent samples ANOVA versus the repeated measures ANOVA.*0551

*People have the same start, they have the same hypothesis not only that but they both have the *0561

*same idea of taking all the variance in our sample and breaking it down into component parts.*0567

*Now what we talk about all the variance in our sample we really mean what is our sum of squares total.*0572

*What is the total amount of variability away from the grand mean in our entire data set? *0581

*And we can easily just from the sentence we could figure out what the formula for this would be.*0590

*This should be something like the variability of all every single one of our data point minus the *0597

*grand mean which we signify with two bars the double bar square and the Sigma now to do this *0606

*for every single data point not just the data point in one sample while the way it knows to do that is because this should say N total.*0616

*So this is going to go through every single data point in every single sample and subtract get the *0625

*distance from the grand mean in the square that distance and add those distances all squared.*0634

*Okay so that is the same idea to begin with.*0643

*Now we will take this at this total and break it down into its component parts.*0647

*Now an independent samples what we see is that all of the variability that were unable to *0651

*explain lies within the group, all of the variability that we are very interested in, that is between *0664

*the group, and so independent samples the story becomes, as this total is a conglomeration, it is *0671

*when you split it up into its part you see that it is made up of the sum of squares within the *0681

*group inside of the group and the sum of squares between the groups added up.*0690

*And because of that the S statistic here becomes the variance between over the variance within *0696

*and obviously each of these variances corresponds to its own sum of squares.*0711

*Now the repeated measures ANOVA were going to be talking about something slightly different because now we have these linked data.*0720

*So here the data is independent these samples are independent they are not linked to each other in any way.*0730

*Here, these samples are actually linked to each other.*0738

*Either by virtue of being made from the same subject or the same class produced or something about these scores are linked to each other.*0744

*So not only is there variability across the groups just like before so sort of between the groups and variability within the group.*0753

*But now we have a new kind of variability.*0770

*We have the variability caused by these different linkages, these all are different from each other but maybe similar across.*0774

*So the person who owns a digital camera they might just have an enormous number of photos all across the board.*0785

*The person who does not have a digital camera or not even a smartphone might have a low number of photos across-the-board.*0792

*So there are those things that we call often are called individual differences.*0799

*Those are differences that we actually mathematically quantify, we could actually explain where *0806

*it is but we are not actually interested in the study, were really interested in the between group difference.*0813

*But this is not all.*0822

*Once you have taken out this individual of variability there is still some residual within group variability left over.*0825

*And so that is that is really stuff we cannot explain, it is not caused by the individual differences *0834

*it is not because of between group, it is just within group differences.*0842

*So in repeated measures the sum of squares total actually breaks down slightly differently even *0847

*though it is still it is still this idea of breaking down the sum of squares total now it actually splits *0856

*up into some of squares subjects, this individual links the yellow part plus the sum of squares *0865

*within just like before that now we call it residual because this we have taken out the sum of the *0878

*variability that comes from the individual differences and so because of that there is there is only *0889

*left over and so because that we call it residual just like the words left over.*0896

*And of course the sum of squares between which is what were actually very interested in.*0902

*So just to recap, this is something that we can explain how there were not interested in, this is *0907

*something we cannot explain and this is something we are very interested in.*0915

*So, our S statistic will actually become our variability between divided by our variability residual *0920

*and in fact we just wanted to take this guy out of the equation we want him to out of the equation of F.*0931

*So F, it does not count the variability from the subjects, were individual difference, the are not interested in that.*0940

*Okay so I wanted to show you this and a picture here is what I would show you.*0953

*Here is what we mean by independent samples, remember, the independent samples ANOVA it *0960

*is always been a start off with that same idea and total the difference between each data point *0966

*from the grand mean squared and then add all of those up that is the total sum of squares.*0975

*In independent samples, what were going to do is take all of this all of this variability that SS total.*0983

*That is that SS total of the total variance and we are going to break them up into between group variance.*0994

*So think of this, this is just to signify the difference of all of these guys from the grand mean.*1012

*So the between group differences, so SS between and add to that the within group variability.*1020

*The variability that we have no explanation for.*1038

*So that is the within group variability.*1041

*So it only makes sense that the variability between divided by the variability within, this is what *1046

*we would use in order to figure out the ratio of the variability we are interested in or *1067

*hypothesizing about divided by the variability we cannot account for.*1077

*So this becomes the S statistics that were very much interested in.*1090

*Now when we talk about the repeated measures ANOVA, once again we start off similarly for *1096

*every single data point we want their squared distance away from the grand mean and add them all up.*1106

*In order to see this as a picture you want to see that whole this whole idea here that did the *1113

*distance of all of these away from the grand mean that is SS total.*1123

*However what we wanted to do is then break it up into its component parts and just like before *1129

*we have these differences between the groups so that is SS between.*1138

*And that SS between is the stuff that were really interested in so that is also going to be a factor here.*1147

*Where we take the variability between but then, we want to break up the rest of the variance *1156

*into one part that we can actually explain, we could account for it and into the rest of the residuals that we cannot explain.*1165

*So even when we are not interested in that we could actually account for the variability.*1174

*You could think of it across these rows because notice that person one, the viewer photos *1182

*across-the-board, person three just has more photos across-the-board and so those are the kinds *1194

*of individual differences, little level differences that we do not actually want in our S statistic.*1204

*Its variability we know where it comes from which is not interested in it in terms of our hypothesis testing.*1211

*So we have this SS subject, put a little yellow highlight here so that you know what it stands for *1217

*and that is the variability that we can explain but not part of my hypothesis testing.*1232

*And so what variability are we left with, we are left with any leftover variability, there is some *1240

*leftover variability and we call that residual variability and that is going to be SS residual.*1248

*And if we want to look at the variability that were interested in over the variability we cannot *1257

*explain, we are not going to include this variability, we are only going to use this one.*1264

*So the variability between groups divided by the variability residual, residual variability.*1269

*Once we have that now let us break it down even further.*1281

*So the repeated measures S statistic now you sort of know basically what it is.*1289

*It is the is the variability between groups divided by the variability within group.*1294

*Now we could break these up into their component parts so it is going to be the SS between *1307

*sum of squares between, divided by the degrees of freedom between, all divided by the sum of *1316

*squares of partnership residual, residual over the degrees of freedom residual.*1327

*So far it just looks like what we have always been doing with variability sum of squares through *1338

*the freedom but now we have to figure out okay how do we actually find these things.*1348

*And in fact, this is something you already know because this is actually exactly the same as the independent samples ANOVA.*1355

*The only thing that can really be different is this one.*1371

*Okay so let us start off here.*1387

*So this is what we are really looking for when we start double-click on this guy we double-click *1390

*on that variability what we find inside is something like this and then double-click on each of these things and figure out what is inside.*1398

*So conceptually this is idea the whole idea of the variability between groups is the difference of *1405

*sample mean from the grand mean because we want to know how each sample differs from that grand mean.*1413

*Now let us think about how many means we have because that is going to determine our degrees of freedom.*1419

*So how many means we have is usually K.*1426

*How many samples we have and so with the degrees of freedom between well it is going to be K -1.*1430

*And the way you can think about this is how many means we find, where you find three means *1441

*or how many means as groups so if we have four groups it would be for means if we had three groups it would be three means.*1448

*And if we knew two of them and we know the grand mean, we could actually figure out third and *1456

*so because of that is going our degrees of freedom is K – 1, the number means -1.*1465

*Okay so what is sum of squares between? *1471

*Well it is this whole idea of the difference of sample means from the grand mean and we could *1479

*say that the sample mean away from the grand mean we have a whole bunch of sample means.*1484

*Something to put my index there.*1489

*And we are going to square that because the study some of squares, and each means distance *1491

*should count more if that sample has a lot of members and I should get more votes so we are *1498

*going to multiply that by N sub I, how many in their sample.*1504

*And in order to figure out what I mean when I say okay let us think about that, I is going to stand *1511

*for each group so this Sigma is going to have to go from I = 1 through K how many groups.*1518

*And then it is going to cycle through group 1, group 2, group 3, group 4 and this is some of squares between.*1527

*Additional is very similar because this is actually the same thing from independent samples ANOVA.*1536

*So now we have to figure out how to find the other sum of squares the new one.*1544

*Sum of squares residual and degrees of freedom residual and the whole reason we want to do *1551

*that is because we want to find the variability residual, the leftover variability.*1555

*If any leftover spread within the groups is not accounted for by within subject variation.*1562

*Now within subject might mean within each person right but it might mean within each hamster *1570

*or each company that is being measured here repeatedly so whatever it is that your case is *1578

*whether animal or human or entity of some sort , that is considered within your subject of variability.*1585

*And those subjects are all slightly different from each other.*1595

*But that is not something that were actually interested in so we want to take that out and take the leftover variability.*1598

*And because it is the idea of leftover, we actually cant find a lot of this, we can find some of *1604

*squares directly, we have to find the leftover.*1614

*And so the way we do that is take the total sum of squares and then subtract out the stuff we do *1617

*not need which is namely the sum of squares between as well as the sum of squares within subject to the variability within subject.*1624

*And so here we see that we are going to have to find some of squares for everybody were *1637

*enough to find total as well as for the within subject, we already knew we have to find this one, *1642

*and that is how where we can find some of squares residual, literally whatever is left over.*1650

*In the same way to find degrees of freedom residual, we should have to know something about *1656

*the other degrees of freedom in order to find this sort of our whatever's left.*1661

*And so in order to find degrees of freedom residual, what we do is we multiply together the *1667

*degrees of freedom between times the degrees of freedom within subject and when we do this, *1674

*we are going to be able to find all the degrees of freedom that is leftover .*1683

*Okay so we realize in order to find sum of squares residual we have to find all these other sum of squares so here is some of squares within subject.*1689

*So the way to sort of think about this notion is this idea that were really talking about subject level or case level, subject level variation.*1702

*So each case differs a little better from the other cases for God knows what reason right but we *1716

*can actually account for it here, it is not totally unexplained we do not know why it exists, we *1723

*know it exists because the subjects are all slightly different from each other and we do not know *1729

*why it exists but we know what it is and we could calculate it.*1734

*Okay so conceptually you want to think about this at how far each subjects mean is away from the grand mean.*1738

*Remember in repeated measures we are repeatedly measuring each subject or case, we are *1748

*measuring them multiple times so if I am Facebook user I will be contributing for different scores to this problem.*1754

*Now I know what you could do is get a little mean just for me right? *1763

*The little mean of my four scores and that is my subject mean.*1770

*So each subject has her own little mean and we want to find the distance of those little means away from the grand mean.*1774

*So let us think how many subject means do we have? *1783

*We have N number of subjects, that we have an number of samples for each for each sample and number of measures for each sample.*1787

*So that is our sample size.*1800

*So what is degrees of freedom for within subjects? *1803

*Well, that is going to be N-1.*1806

*So what is the sum of squares for each subject? *1808

*Well one of the things you have to do is sort of figure out a way to talk about the subject level mean.*1814

*So here, I am just going to say mean and put a index for now but here in my in my little telling that *1820

*this Sigma will tell this what I is, I will go from one up to N sample size.*1835

*This is really the subject means and I want to get the distance squared distance from each *1843

*subject mean to the grand mean and square that, squared distance and we should also take into *1853

*account how many times is the subject being measured and that is going to be K number of times.*1860

*How many how many samples are taken how many measures are taken so repeated measures how many times the measure is repeated.*1867

*And the more times a subject participates the more this variation will count.*1878

*So there we have subject level variation, we are really only finding it so that we can find SS *1889

*residual, so better do it and so we also have to find sum of squares total and degrees of freedom total.*1899

*These are something we have gone over that just to drive it home remember the reason we *1908

*want to find this is just so we can find sum of squares residual.*1913

*So conceptually this is just the total variation of all the data points away from the grand mean.*1916

*What is the total number of data points? *1922

*That is going to be N total.*1925

*So every single data point counted up and the way we find that is sample size N times the *1928

*number of samples we have so if we had 30 people participating in four different measures, it is *1937

*30 times 4 and the number of samples is called K so NK of N subtotal.*1944

*So what is the degrees of freedom total? *1954

*Well it is either going to be N total minus 1 or the same exact numerical value will be NK -1 either way.*1957

*And so what is the sum of squares total? *1967

*Well we have already been through it this is what we always start off with at least conceptually *1970

*for every single data point notice that there is no bars on it not any means of literally every single data point.*1975

*The distance from the grand mean squared and we could put NK here just to say go and do this *1984

*for every single data point do not leave one behind.*1994

*So we have all of our different components, now let us put them together in this chart so that you will know how they fit together.*1997

*Remember the idea of the F is the variation we are interested in over variation we cannot *2009

*explain, we cannot account for do not know where it comes from, it is a mystery.*2027

*The formula for this is going to be F equals, I remember this is for repeated measures so that *2033

*between sample variability over the residual variability.*2043

*And in order to find that we are going to need between sample variability.*2051

*The idea is always going to be the best sample means difference from grand mean.*2057

*So basically the centres of each sample away the distance away from the grand mean and so the *2072

*formula for that is going to be S squared between equals SS between over the DF between and *2084

*we can find each of those component parts SS between going to be the sum to many zigzag, the *2094

*sum of all of my all of my X bars minus the grand mean the distance and when I say all I mean one at a time.*2103

*Each as individuals one at a time, and this distance should count more if you have more people *2118

*or more data point in your sample and I does not go from one to N, it goes from one to K, I am *2125

*going to do this for each sample, NK is my number of samples or number of groups.*2134

*So my degrees of freedom between is really going to be K -1, number groups -1.*2139

*Okay so now let us try to get residual variability.*2148

*Now residual variability is that leftover within groups within sample variability, now in order to *2155

*get leftover the formula for this is going to be the variability residual.*2169

*Now to get that you get the residual sum of squares and divide by the residual degrees of *2181

*freedom, the residual sum of squares is literally going to be the left over.*2190

*SS total minus SS subject plus SS between.*2197

*And my degrees of freedom residual is going to be a conglomeration of other degrees of *2209

*freedom available to S times the degrees of freedom between, okay.*2222

*So we know that in order to find these, so the total variability let us start there, we know this one *2230

*pretty well, all the data point in all of our samples away from the granting and so we actually do *2242

*not need the variability here and we do not need this variability either.*2269

*What we really need is the sum of squares total and that is going to be for each data point no X *2274

*bar anything get this squared distance from the grand mean.*2283

*So now that we have that we do not really need that but we can find it anyway so the degrees of *2288

*freedom total is going to be NK -1 so the total number of data points minus 1.*2299

*Now let us talk about within subject variability this is the Brad of each case away from grand *2308

*mean and when you talk about each case, each case can sort of be represented the point *2324

*estimate of it can be its own means so each cases mean so that is how I want you to think of it.*2331

*Each case is represented by its own little mean and so that is why they were using it means to calculate the distance.*2337

*So that SS subject is going to be the distance of each subject level mean away from the grand *2345

*mean squared and in order to say subject level, you got to put that N here so that it knows do *2360

*this for each subject not do this for each data point or do this for each, if we put a K there would *2368

*be do this for each group and we wanted to count more if they participate in more measures so if the measures are repeated over and over again.*2386

*So we want to put in the number of k and so that gives us our are some of squares for each *2387

*subject and once we have those two we can find this as well as some of squares between and *2394

*then we also need the degrees of freedom for within subjects just because were in need that to find out the degrees of freedom residual.*2400

*This guy do all this jump through hoops.*2411

*So the degree of freedom for each subject for subject level variance is going to be N -1, the number of subjects -1.*2414

*Okay so here is example 1 which is more prevalent uploaded mobile profile photos and so these *2423

*are all different kinds of photos but one person or one Facebook user presumably 1 person, they *2434

*are sort of the linking factor of all four of those measures.*2441

*So what is the null hypothesis? *2448

*Well it is that all of these groups really come from the same population.*2451

*The reason I use this P notation is for just different types of photos and I will call this one 2, 3, and 4.*2457

*Also it makes it easier for me to write my alternative hypothesis, it has a practical significance so all use of P’s are not equal so they are not all equal.*2472

*So the significance level we could just set it as alpha equals .05 just by convention because we *2494

*are going to be using F value, we do not have to determine whether the one tailed or two-tailed.*2513

*Always one tailed is cut off on one side and skewed to the rights it is always going to be just on *2517

*the positive side and so let us draw our decision stage with the jar of distribution.*2528

*We know that it ends at zero is alpha equals .05, what is the F here? *2535

*Well remember, in order to find as we need to know the denominators DF as well as this numerators DS.*2548

*And so here we know that F in the numerator is going to be degrees of freedom between group and that is K -1.*2557

*There is 4 groups so it is going to be 3 and the degrees of freedom of residual is going to be degrees of freedom between times degrees of freedom subject.*2570

*So we are going to need to find degrees of freedom subject and degrees of freedom subject is going to be N-1.*2587

*Now let us look at our data set in order to figure out how many we have in our sample.*2597

*So I have made it nice and pretty here, first type photos mobile uploads uploaded photos and *2602

*profile photos, as you look at this row it has all of the data from one subject so this *2609

*person has zero photos of any kind whereas let us look at this person.*2619

*This person has zero mobile uploads and zero profile photos that they have 79 uploaded photos *2625

*and 37 tag photos and so for each subject we can see that there is some variation there but *2631

*across the different samples we also see some variation.*2639

*So here down here I put step one they are all equal and are not all equal, equals .05, here is the *2643

*decision stage, our K is 4 groups 4 samples, our degrees of freedom between us 4 -1, we already *2655

*done that but might to fill this in, on degrees of freedom for subject the reason why this is there *2663

*is so that we can find the degrees of freedom residual and once we have that then we can find our critical F.*2670

*So the degrees of freedom for each subject we should count how many subjects we actually have *2692

*here, we could just count the rows, so I just picked profile photos -1 so we actually have 29, 29 *2707

*cases but our degrees of freedom for subject is 28.*2719

*Now degrees of freedom residual are those 2° of freedom multiplied to each other so 3×28 and *2724

*that is going to be 84 and that is our denominator degrees of freedom.*2735

*So now we can find our critical F.*2740

*In order to do that we use F inverse the probability is .05 and our first degrees of freedom is the *2742

*numerator one and our second degrees of freedom is the denominator and our critical F is 2.71, that is our critical F.*2750

*So once we have that we can now go on to sort of figure out, okay from there let us go on and calculate our sample tests.*2767

*So we will have to find the sample S statistic right before, I disputed generically because you *2777

*might have to find T statistic or the statistics but in that case we know because we have a *2787

*omnivorous hypothesis we need that S statistic and we have to find the P value afterwards so let us find the S statistics.*2794

*Go to your example again, this is example 1, let us put in all the different things you need.*2803

*So you need the variance between over the variance of the residual variance so let us start off *2814

*with variance between, it is something we already know, we know it is been split up into sum of *2822

*squares between and degrees of freedom between.*2827

*We actually have degrees of freedom between already so let us just fill that in, in order to find *2830

*the sum of squares between, you have to find the means for each of these groups.*2834

*We are also going to need to find out what is the N.*2841

*That is actually quite simple because we know that it is 29 for each of these groups so that makes life a little bit simpler.*2845

*Now let us find the averages for each of these samples so for the first sample I believe this is tag photos, the mean is 9.93, *2861

*I believe this is mobile uploads, that is 12.45 for uploaded photos, that averages 68 and finally for profile photos the average is 1.5.*2874

*Okay so now we going to have to calculate the grand mean.*2905

*The grand mean is quite easy to do on XL because you just take all your data points every single one and you calculate that average.*2909

*The average is 23.*2919

*I am just going to copy and paste that here, what I did was they put a point here so that it would *2921

*just point to that top value for the granting shouldn't change the granting is always the same.*2929

*Now that we have all of these values we could find N times XR minus the grand mean squared.*2935

*We could find that for each group, and then when we add that up, we end up getting our sum of *2948

*squares between and we get this joint number 82,700.*2968

*And so I am just going to put a pointer = point to that guy and then I am going to find out my variance between.*2973

*So my variance is still quite large about 27,600.*2986

*Okay so to have that now we need to find my variance of my residual variance.*2993

*In order to find residual variance, I know I am going to need to find all this other stuff that I did not necessarily plan on.*3000

*So one of the things I do need to find is my SS total as well as my SS subject.*3007

*I am going to start with SS total because although the idea is simple to on XL it looks a little crazy *3014

*just because it takes up a lot of space because we going to need to find this square distance *3023

*away from the grand mean for every single data point.*3028

*So here, all my data points are here.*3033

*Now I am going to need to find the square distance of this guy away from the grand mean, and then add them all up.*3040

*What is helpful in XL is to create separate rows and then to sort of add them up and so I am just *3055

*going to use save these for later, and so this is, I have already put in the formulas here, this 1 is *3067

*tag for the tag photos, it is sort of my partial way to find SS total just for the tag photos and I am *3075

*going to do it for the mobile photos and for the uploaded photos then for profile photos and then add them altogether .*3085

*So either sort of subtotal.*3091

*So what I need to find is that data points minus the grand mean and I will just use this grand mean that I found down here.*3094

*But what I need to do is I need to lock that down I need to say always use this grand mean do not use any other one.*3105

*You put that in parentheses so that I could square it.*3113

*So here I am going to do that all the way down for tag photos and just take this across for mobile *3118

*uploaded and profile photos and that is the nice thing about XL it will give you all of these values very very easily.*3133

*I am just going to shortness this for second, just to show you what each of these is talking about.*3144

*So click on this one, this cell gives me this value minus my grand mean which is locked down *3152

*squared so I have now and every single data points square distance away from the grand mean and these are all the differences square distance.*3160

*Now I need to add them all up.*3171

*So put sum, and I am not just going to add up this column I am literally going to add all this up.*3174

*So our total sum of squares is 257,000.*3184

*So I am going to go down to my sum of squares total and just put a pointer here and say that is it.*3192

*So how do I find my sum of squares for the subject level variation? *3201

*Well, this I know I need to find the mean for every subject then I need to find the distance *3212

*between that mean and the grand mean square that and multiply it by how many groups I have.*3219

*The nice thing is the number of groups I have this constant is always four for everybody so let us go ahead and find subject level mean.*3226

*So subject means are going to be found by averaging one person measures for all 4 sample and *3235

*so that guys average of zero, just copy and paste that down, if they wanted to check this one takes the average of these four measures.*3248

*So this is subject level variation and this shows you that this guy has a lot fewer photos period than this guy.*3259

*He has just an average a lot higher photos than this guy.*3268

*And this guy is sort of in the middle of those two.*3273

*Once we have these subject level means now we could find this idea K times the difference *3276

*squared for each subject so I know my K is going to be 4 times my subject level mean minus the *3286

*grand mean and I will just use my already calculated grand mean down here and I need to lock *3302

*that grand mean down because that grand mean is never going to change squared.*3309

*Once I have that then I probably want to add them all up in order to get my sum of squares for within subject variation.*3316

*I will just put this little sum signs so that I know that this is an another like data point, it is a *3335

*totally different thing, sum, and once I have that it is 56,600 and I know my sum of squares within subject.*3345

*Once I knew all those things now I can finally calculate some of squares residual because I know my ingredients.*3360

*I have my sum of squares total minus the sum of squares per subject plus the sum of squares *3369

*between and I could obviously distribute out that negative sign but I will just use the parentheses.*3380

*So here is my leftover sum of squares that whatever's leftover unaccounted for and I already *3390

*figured out my DF residual and so here I am going to put my sum of squares residual divided by *3399

*degrees of freedom residual and there get 1400.*3410

*So now we can finally finally calculate our F by taking the variance between and dividing that by the variance residual variance.*3416

*In there I get 19.69 which is quite a bit above the critical F of 2.7.*3427

*Now once I have that now I could find my P value.*3435

*So by May P value I would put in my F, put in my F value, my numerator degrees of freedom as *3441

*well as my denominator degrees of freedom and I get 9.3×10 to the negative 10 Power so that *3456

*means there is a lot of decimal places before you get to that 9 so it is very very very very small P value.*3468

*So what do we do? *3475

*We reject the null.*3478

*Also remember that in a F test , all we do is reject the Omnibus null hypothesis that does not *3480

*mean we know which groups are actually different from each other so when you do reject the *3490

*null after doing F test, you want to follow up and do post hoc test.*3495

*There is lots of different post hoc test you might learn to keep postop or Bonferroni corrections *3500

*so those all help us know the pairwise comparisons to figure out which means are actually *3507

*different from which other means and you probably also want to find effect size and in F test if *3515

*effect size is not D or G instead its Eta squared.*3524

*So we would reject a null.*3527

*Example 2, a weightless boot camp is trying out three different exercise programs to help their clients shed some extra pounds.*3537

*All participants are assigned to team up 4 people and each week their entire team is weight *3546

*together to see how many pounds they were able to take off.*3552

*The data shows their weekly weight loss as a team.*3554

*With a exercise program all equally effective in helping them lose weight note that all teams *3558

*tried all three exercise regime but they all receive the treatment in random order.*3564

*So this is definitely a case where we have three different treatments.*3569

*Treatment 1, 2 and 3 and we have data points which are going to be pounds lost.*3574

*How many pounds they were able to take off per week pounds loss per week but these are not independent samples.*3581

*They are actually linked to each other.*3590

*What's the link? *3592

*It is the team of four that lost that weight right so this team lost that much under this exercise *3594

*regime, lost that much under this exercise regime, lost that much under this exercise regime.*3602

*Now each team got these three exercise regimes in a different order.*3608

*Some people are 3, 2, 1, so they have all been balanced in that way so if you pull up your examples and good example 2, you will see this data set.*3612

*So here are the different teams or squads.*3627

*Here are the three different types of exercise program and in the different orders that they *3630

*were, that they did these exercises and each exercise was done for a week.*3635

*So let us think about this.*3642

*So to begin with we need a hypothesis so step one is the null hypothesis and all are equal.*3644

*So all the mutinies, exercise 1, exercise 2, exercise 3, they are all equal.*3660

*The alternative hypothesis is that not all are equal.*3667

*So step 2 is our significance level we could just set alpha equals to .05 once again because it on *3674

*the best hypothesis we know we are going to do a F test so it does not need to be two-tailed.*3686

*So step three this is the decision stage if user imagine that F distribution or color in that part, *3692

*what is that critical F? *3703

*Well, in order to find the critical F, we are going to need to find the DF between as well as the DF *3706

*residual because that is the numerator and the denominator degrees of freedom.*3715

*In order to find DF residual we also need to find DF subject and remember here subject does not *3722

*mean each individual person, subject really mean case.*3730

*And each case here is a squad.*3733

*So how many squads are there -1.*3736

*So count how many squads there are -1.*3741

*So there is 11° of freedom or subject.*3746

*For degrees of freedom between what were going to need is the number of different samples which is three okay -1 so 3 - 1 is 2.*3760

*And so my DF residual is the DF between times the DF subject and that is 22 so let us find the critical F.*3774

*We need F inverse, the probability that we need is .05, the degrees of freedom for the *3784

*numerator is 2, the degrees of freedom for the denominator is 22 and our critical F is 3.44.*3792

*Step 4, here we are going to need the F statistic and in order to find F, we need the variance between divided by the variance residual.*3802

*In order to find variance between we are going to need the SS between divided by DF between, *3823

*we already have DF between thankfully, so we do need SS between.*3833

*And the concept of SS between is the whole idea of each samples X bar, their distant away from *3838

*the grand mean squared and then depending on how many subjects you had in your sample how *3849

*many data points you had in your sample you get waited more or less.*3856

*Now the nice thing is all of these have the same number of subjects.*3860

*But let us go ahead and and try to do this.*3864

*So first we need the different samples so this exercise 1, exercise 2, exercise 3, we need their N, *3869

*their N is going to be 12, there is 12 data points in each sample.*3879

*We also need there each exercise regimes average weight loss so we need X bar and we also *3888

*need the grand mean because ultimately we are going to look for N times X bar minus the grand *3901

*mean squared in order to add all of those up.*3908

*So let us find X bars for exercise regime number 1.*3912

*So that an XL makes it nice and easy for us to just find out all those averages very quickly and *3918

*then once we have that, now we can find the grand mean.*3931

*The grand mean is also very easy to find here.*3938

*We just want to select all the data points.*3941

*I think I selected one of them twice, be careful about that.*3944

*So make sure everybody is selected just one time so this is the average weight loss per week *3950

*regardless of which team you were on regardless of which exercise you did.*3960

*And now let us find N times the X bar minus the grand mean squared and let us do that for each for each exercise regime.*3965

*Once we have that done we could find the sum, and the sum is 23.63.*3983

*So here in SS between I would put that number there.*3997

*So once we have that now we could actually find this because we already have calculated the DF between, was not too hard.*4006

*Now we have to work on variance residual, now in order to find variance residual, let me just add *4018

*in a couple of rows here just to give me a little more space, variance residual, in order to find *4031

*variance residual I am going to need to find SS residual divided by DF residual.*4049

*We already have DF residual so we just need to find SS residual, in order to find that I need SS *4054

*total minus SS between + SS subject level.*4062

*So I already have my SS between so I need to find SS total and SS for each subject.*4071

*So SS total is going to be for every single exercise regime, for every single one of these data *4080

*points I need to find that distance away from the grand mean, add them all up and square and that is going to be my SS total.*4092

*So for E1 here is my subtotal for SS total, for E2, my subtotal for SS total, for E3, my subtotal for SS total.*4104

*So that is X minus the grand mean, lock that grand mean down, squared and make sure you do *4120

*that for every single data point in E1 so if I check on that last data point and just go ahead and *4141

*copy and paste that although it have to hear let us just checked on this one, this is taking this *4151

*value, subtracting the grand mean from it and then squaring that distance.*4157

*So once I have this, I could sum them all up and get my SS total, my total sum of squared distances.*4164

*So I am just going to put a pointer here so that I do not have to rewrite the number.*4180

*Once I have that all I have to find the SS subject.*4187

*Now remember, the SS subject each subject has its own little mean could be repeatedly make *4190

*the measure right so we have to find the subjects mean and then we have to get the distance *4195

*between their mean and the grand mean, square that and multiply it by the number of measures, K.*4201

*So let us do that here, first we need to find the subjects X bars so that is going to be each squads *4211

*average weight loss so some squads probably lost more weight than others, so this is the average *4226

*weight loss for each squad so it looks like you know this squad loss a bit, so a little bit of variation *4242

*in subjects success and sure we are going to look at K times the subjects X bar minus the grand *4259

*mean squared so we already know K, K is going to be 3 times the subjects X bar minus the grand *4272

*mean, I am just going to use the one we have already calculated down here and of course lock that down so copy and paste this, squared.*4284

*So copy and paste that all the way down and I could find the summary here and this is going to be my sum of squares for subject.*4298

*That is the sum of the bunch of squares.*4312

*So that is 34 something.*4318

*I am just going to put a pointer there so I do not have to retype that but I could just see it nice and clearly right here.*4321

*So now you have everything I need in order to find SS residual so I need SS total minus my sum of squares between plus some of squares subject.*4332

*Once I have that now I could find my vary residual variance divided by degrees of freedom, okay *4344

*so here it looks like my residual variance is much smaller than my between sample variance and *4356

*so I could predict my F value will be pretty big so 11 point something divided by two point *4372

*something and that gives me 5.219 and that is a little bit bigger than my critical F.*4381

*So if I find my key value F disc and put in my F, my numerator degrees of freedom, my *4391

*denominator degrees of freedom, I would find .01 so that seems like a pretty small, smaller than *4403

*.05 so I am going to be rejecting my null.*4414

*So step five down here, reject the null.*4418

*And we know that once you reject the null you are going to need to also do post hoc tests as well as find it a square.*4425

*So that brings us to example 3 what is the problem with a bunch of tiny t-test? *4432

*Well with so many t-test the probability of type 1 error increases increasing the cut off A, actually *4445

*were not increasing the cut off, we are keeping it at .05 but the type 1 error increases because *4458

*we reject we have the possibility of rejecting the null multiple times.*4466

*With so many t-test the probability of type 1 error increases here it is because we may be rejecting more null hypothesis.*4470

*This is actually a correct answer so we might not be done yet.*4478

*With so many paired samples t-test we have a better estimate of S because we have been estimating S several times.*4484

*With so many paired samples t-test we have a poor estimate of S because were not using all of *4491

*the data to estimate one S in fact we are just using substance of the data to estimate S several times, that is a good answer.*4500

*So that is it for repeated measures ANOVA, thanks for using educator.com.*4508

*Hi, welcome to educator.com. *0000

*We are going to talk about the chi-square goodness of fit test. *0002

*So first, we are going to start with the bigger review of where the chi-square test actually fits in. *0005

*Amongst all the different inferential statistics we have been learning so far and then we are going to talk *0012

*about a new kind of hypothesis testing, the goodness of fit hypothesis test. *0018

*So it is going to be similar to hypothesis testing as we been doing so far but there is a slightly different logic behind it.*0023

*So because it is a slightly different logic there is a new all hypothesis as well as the alternative hypothesis. *0029

*Then we are going to introduce the chi-square distribution and the chi-square statistic. *0037

*And then we are going to talk about the conditions for chi-square test when do we actually do it. *0044

*So where does the chi-square test belong? *0049

*And it is been a while since we have looked at this if you are going in order with the videos but I think it is*0054

*pretty good to stop right now and sort of think where we come from? *0059

*Where are we now? *0063

*So the first thing we want to think about are the different independent variables that we been able to look at. *0065

*We been able to look at independent variables the predictor variables that are either categorical or continuous. *0072

*When the idea is categorical you have groups right? *0084

*Or different samples, right? *0095

*When the idea is continuous you do not have different groups you have a different levels that predict something. *0098

*So just to give you a idea of a categorical IV that would be something like experimental group versus the *0107

*control group or something like this categorical IV may be someone who gets a drug versus someone who *0116

*gets the placebo , a group that gets the drivers of the group that gets the placebo and example of the *0127

*continuous IV might be looking at how much you study predicting your score on a test , so how much you *0132

*study would be a continuous IV. *0140

*So that is one of the dimensions that we need to know, is your IV categorical or continuous. *0143

*You also need to know whether the DV is categorical or continuous so the DV is the thing that were *0150

*interested in measuring at the end of the day the things that we want to know that this thing change this is*0160

*the thing we want to predict right, and so far here is how would come. *0167

*At the very beginning we looked at continuous types of tests and those types of measures and those were *0177

*the regression, linear regression, as well as correlation. *0187

*Remember R and regression was that stuff about like Y equals the not + b sub 1 times X, so that was *0193

*regression and correlation way back in the day. *0210

*We have been covering a lot of this quadrant actually looking at t-tests and ANOVA right?*0215

* One important thing to know that t-tests and ANOVAs are both hypothesis tests, only so far have not *0224

*learned hypothesis testing with regression and correlation. *0238

*A lot of inferential statistics in college does not cover hypothesis testing of regression until you get to more advance levels of statistics. *0241

*So what do ANOVAs and t-tests sort of have in common? *0255

*Well they have in common that they are both categorical IV and continuous DV. *0261

*The IV is categorical and you only have one, one IV. *0269

*And your DV is continuous. *0277

*So that sort of what they have in common, what is different about them? *0282

*Well the difference is that the IV in t-tests has two levels in only two levels so there is only two groups or two samples. *0287

*In ANOVAs we could test for more than two samples, we can do that for 3 4 5 samples. *0297

*So that IV has greater than two levels and so that is where we been spending a lot of our time. *0302

*So for the most part continuous DV are really important because they tell us a lot, they tell us the find ways*0312

*that we could actually be different, that the data could actually be different.*0320

*So you are going to, it is more rare that you will use the categorical dependent variable, that is not going to*0327

*be as informative to us but it is still possible and that is where the chi-square is going to come in. *0334

*The chi-square is been coming right in this quadrant where we have categorical IV also a categorical DV so *0340

*for instance we might want to see something like if you are given a particular job or the placebo, do you *0347

*feel like you are getting better, yes or no right? *0357

*So that is a categorical DV, it is not like the score that we can find a mean and so this is where the chi-square tests come in. *0360

*And there is going to be 2 chi-square tests that we are going to look at. *0375

*The first one, we are going to cover today and it is called goodness of fit. *0379

*The next one is in the next lesson and it is called a test of homogeneity. *0382

*They are both chi-square test. *0386

*The other way you will see that what is written is chi-squared, so sometimes, do not think of, oh what is this doing here? *0387

*When it has this little curvy part here we need chi-square, the Greek letter chi, finally this is a test that*0398

*rarely is covered in inferential statistics but at more advanced levels of statistics he did cover it and it is called *0407

*the logistic test and logistic test takes you from continuous IV to categorical DV. *0415

*But that is rare design used in conducting science, it is not as informative as continuous to continuous or categorical to continues. *0424

*Alright so we are going to spend your time right in here. *0436

*So there is a new twist on hypothesis testing, it is not totally different, it is still very similar but there is there is a subtle difference. *0441

*Today we are going to start off with the chi-square goodness of fit test. *0454

*Basically let us think about hypothesis testing in general. *0457

*In general you want to determine whether a sample is very different from expected results that is the big idea of hypothesis testing *0462

*and expected results come from your hypothesized population.*0470

*If your sample is very different than we usually determine that with some sort of test statistic and looking*0474

*at how far it is on the on the tested statistics distribution right and we look at whether it is past that Alpha*0481

*cut off or the critical test statistic right and then we say, oh this sample is so different than would be *0489

*expected given that the null hypothesis is true that we are going to reject the null hypothesis. *0496

*That is usually hypothesis testing. It still takes that idea whether to look at whether a sample is very *0504

*different from expected results, but the question is how are we going to compare these two things? *0511

*We are not going to compare means anymore, we are not going to look at the distance between means, *0517

*nor are we going to look at the proportion of variances that is not what we are going to look at either. *0521

*Instead we are going to determine whether the sample proportions for some category are very different *0527

*from the hypothesized population proportion. *0539

*And the question will be how do we determine very different and here is what I mean by determine *0542

*whether the sample proportions are different from the hypothesized population proportion.*0549

*So here I am just going to draw for you sort of schematically what the hypothesized population proportions might look like. *0554

*So this is just sort about the idea, so you might think of the population as being like this and in the *0569

*population you might see a proportion of one third being blue, one third being red, and one third being yellow. *0577

*Now already it is hard to think about like you could already sort of see, well we cannot get the average of*0588

*blue red and yellow right like what would be the average of that, and how would you find the variability of *0597

*that so already we are starting to see why you cannot use t-tests or ANOVAs if you cannot find the mean or *0605

*variance you cannot use those test so is this is what our hypothesized population looks like and when we *0613

*get a sample we get a little sample from that population, we want to know whether our sample *0622

*proportions are very different from the hypothesized proportions or not, so let us say in our sample *0631

*proportion we get mostly blue, little bit of red, little bit of yellow so let say 60% blue 20% red 20% yellow. *0637

*Are those proportions different enough from our hypothesized proportion?*0650

*Another sample we might get is you know, half blue and half red and no yellow, is that really different from our hypothesized proportion? *0655

*Another sample we might get might be only like 110 blue and then 40% red and then the other half will be yellow. *0674

*So something like that we want to say if it is really different from these hypothesized population *0694

*proportion, and so that is what our new our new goal is. *0700

*How different are these proportions from these proportion and then the question becomes okay how to *0706

*determine whether something is very different? *0713

*Is this very different or just different? *0717

*How do we determine very different, that is going to be the key question here. *0724

*And that is why we are going to need the chi-square statistic and the chi-square distribution. *0728

*So we are changing our hypotheses a little bit now the null hypotheses is really about proportion and here is what we are talking about. *0733

*The null hypothesis now is that the proportions of the population are real population that we do not know? *0749

*Will this population be like the predicted or theorized proportion and so here we are asking is this unknown*0756

*population like or known population right and it should sound familiar as that sort of the fundamental basis of inferential statistics. *0772

*So that is our new null hypothesis. *0782

*That the proportions in the population are like the predicted will be like the predicted population proportion still be the same. *0785

*Remember sameness is always the hallmark of the null hypothesis alternatively if you want to say at least *0798

*one of the proportion in the population will be different than predicted so going back to our example, if our*0807

*population are hypothesized population is something like one third, one third, one third maybe what we *0816

*will find is something like in our sample will have one third blue but then some smaller proportion like 15% red and on the rest being yellow. *0830

*Now the one third should match up. *0856

*The one third matches up but what about these other two? *0860

*And so an alternative hypothesis at least one proportion in the population will be different from the predicted proportion, *0864

*there just has to be one guy that is different. *0875

*Suggest I give you an example, let us turn this problem into a null hypothesis in an alternative hypothesis.*0878

*So here it said according to early polls candidate A was supposed to win 63% of the votes and candidate B was supposed to win 37%. *0886

*When the votes are counted candidate a won 340 votes while B won 166 votes so here just to give you that *0898

*picture again the null hypothesis population was that candidate A color A in blue, candidate A should have *0908

*won 63% of the vote and candidate B all color in red should have won 37% of the vote so what would be our null hypothesis? *0918

*Our null hypothesis would be that our unknown population will be like this predicted the proportions of my unknown population*0933

*will have the same proportion as our predicted population. *0945

*So here we might see something like A's proportion of votes of the actual real votes should be like this, *0949

*the predicted population, and B’s proportion of votes should be like predicted population. *0982

*So let us say, A’s proportion the real proportion of votes should be like this, and so should B, B should be like this. *1009

*The other way we could say that is that the proportion of votes the real proportion of votes should be like *1017

*the predicted proportion of votes, and then you could just say for every single category for both A and B. *1025

*So what would be the alternative version of this? *1031

*The alternative would say at least one of the proportion one of the categories either A or B one of those *1035

*proportions will be different from the hypothesized proportion. *1043

*And in fact in this example if one of them is different the other will be different to because since we only*1048

*have two categories if we make one really different than the other one will automatically change. *1056

*But later on we might see example 3, 4, 5 category and so in those cases this will make more sense. *1061

*Okay so now let us talk about how to actually find out if out proportions are really off or not. *1070

*Are our proportion statistical outliers are they deviant, are they significant, do they stand out, that is what we want to know. *1080

*And in order to do that we have to use measure called the chi-square statistic instead of the T statistic *1092

*which looks at a distance away in terms of standard error instead of the S statistic which looks at the *1099

*proportion of the variance are interested in over the variance we cannot explain the chi-square does something different. *1106

*It is now looking at expected values what would we expect and what would we actually observe and so the *1113

*chi-square is going to look like this, so be careful that you do not, usually it is like a uppercase accident and*1124

*it is a little bit different than like a regular letter X, it is usually a little more curvy to let you know it is chi-square. *1134

*So the chi-square is really going to be interested in the difference between what we observe the actual *1142

*observed frequency or percentages minus the expected frequency. *1150

*So what were looking at observed versus expected this is what we see in our sample and this is what we *1157

*would predict given our hypothesized population so this is that predicted population part. *1170

*So were interested in the difference between those two frequencies. *1180

*Now although you could use proportions as well you can only do that if you have the same, if you have a *1185

*constant number of items so you probably are safer to go with frequencies because those are assertively *1200

*weeded proportion so you probably want to go with that. *1203

*So were interested in this difference but remember when we look at this different sometimes there can be *1207

*positive sometimes there can be negative and so we what we do here as is usual in statistics as we square *1214

*the whole thing, but we also want to know about this difference as a proportion of what was expected and we want to do this for every category. *1220

*For the number of categories and I goes from one to the number of categories and there is actually an I down here for everything. *1234

*So what this is saying is that for each category, each proportion that you are looking at so in our in our sort*1249

*of toy example with the red blue and yellow, in this example we would do this for blue we would do this *1259

*for red and we would do this for yellow so number of categories, so categories really speak to what are the proportions made of? *1275

*So in here we have three categories so we would do this three times and add those proportions up and we *1291

*want to eventually be able to find observed frequency and the expected frequency. *1315

*Now in the example that we saw with the voting of for candidate A and B, one of the things I hope you *1321

*noticed was that the observed frequencies were given is just number of votes how many people voted but *1330

*the expected frequencies would be expected hypothesized population, that was given as a percentage so *1336

*you cannot subtract votes from percentage, you have to translate them both into something that is the *1346

*same and so in that it is helpful to change the expected percentages into expected frequency and there is *1353

*going to be another reason for changing it into expected frequencies instead of changing the observed *1366

*frequencies into the observed proportion and I am going to that a little bit later. *1371

*So here is what I want you to think of this, is really the square difference between observed and expected *1377

*frequencies as a proportion of expected frequency and you want to do that and you want to sum that over all the categories. *1384

*Once you have that then you get your chi-square value, now let us think about this chi-square value. *1394

*If this difference is very large right so observed frequencies are just very different than expected one, is that difference is very large? *1400

*You are going to have a very large chi-square also if this difference is very small, they are really close to each other, then your chi-square is be very small. *1413

*So chi-square is giving us a measure of how far apart the observed and expected frequencies are, also I *1422

*want to see that the chi-square cannot be negative. *1434

*First of all because were squaring this difference right so the numerator cannot be negative not only that *1439

*the expected frequencies also cannot be negative because we are counting up how many things we have , *1445

*how many things we observed and so this also cannot be negative so this whole thing cannot be negative. *1451

*So already we see in our mind the chi-square distribution will probably be positive and positively skewed *1457

*because it stops at zero there is a wall at zero.*1465

*Okay so now let us actually talk and draw the chi-square distribution so imagine having some sort of data *1470

*set and taking from it over and over again samples so you take a sample and so have this big data set, you *1479

*take the sample and you calculate the chi-square statistic and you plot that. *1487

*And then you put that back in you take another sample and you take the chi-square plotted again and do *1493

*that over and over and over and over again. *1502

*You will never get a value that is below zero and you will get values that might be way higher than zero *1505

*sometimes but for the most part though be clustered over here so you will get a skewed distribution and *1514

*indeed the chi-square distribution is a skewed distribution. *1520

*Now here when we look at this you might think, hey, that looks sort of like the F distribution and you are *1527

*right overall and shape it looks just like the F distribution and in a lot of ways we could apply the reasoning*1536

*from the F distribution directly to the chi-square distribution. *1544

*For instant in the chi-square distribution, our alpha is automatically one tailed it is only on one side and so*1548

*when we say something like alpha equals .05 this is what we mean, we mean that we will reject the null *1556

*when we have a chi-square value that somewhere out here or here or here but we will fail to reject if we *1565

*get a chi-square value in here from our sample. *1573

*Now this chi-square distribution like the S and t-distribution, it is a family of distribution, not just one*1576

*distribution the only one that is just one distribution is the normal distribution. *1586

*The chi-square distribution again depends on degrees of freedom and the degrees of freedom that the chi-*1591

*square depends on is going to be the number of categories -1 . *1598

*So if you have a lot of categories the chi-square it will look distribution will look different if you have a small*1608

*number of the categories like 2, the chi-square distribution will look different. *1615

*So let us talk about what Alpha means here. *1619

*The alpha here is this set significance level we are going to say, we are going to use this as the boundary so*1623

*that if we have a chi-square from our sample that bigger than this boundary then we will reject the null. *1630

*What is the difference now with P value? *1643

*Now the P value said this is the probability so we might have a P value somewhere out here or we might *1647

*have a P value somewhere here, the P value is going to be very similar to other hypothesis test what the P *1656

*value means and other hypothesis test, basically is going to be the probability of getting a high square value*1669

*larger more extreme and in this case there is only one kind of extreme, positive larger than the one from our sample but under condition. *1681

*Remember in this world which one is true? *1700

*The null hypothesis is true. *1703

*So considering if the null hypothesis were true this would be the probability of getting such an extreme chi-*1712

*square value , one that is that large or larger, that is all we need. *1720

*So, in that way the P value is from our data while the alpha is not from our data it is it is just something we sat as the cut off. *1727

*So there are some conditions that we need to know before we use the chi-square. *1737

*When we use the chi-square we cannot just always use it, there are conditions that have to be met so one of the conditions of the chi-square is this. *1745

*Each outcome in the population falls exactly into one of a fixed number of categories, so every time you *1756

*have some sort of case from the population so let us say we are drying out votes. *1765

*Each vote has to fall into one of a fixed number of categories so if it is two candidates, always two *1773

*candidates for every single voter so we cannot compare voters that had two candidates versus voters who had three candidates. *1785

*Also these have to be mutually exclusive categories, one vote cannot go to two candidates at ones so they *1792

*have to be mutually exclusive, you got vote for A or vote for B. *1802

*And you cannot opt out either, or else nobody has to be one of the fixed numbers of categories ahead of time. *1807

*So the numbering is slightly off here but the second condition that must be met is that you must have a *1816

*random sample from your population, that is just like all kinds of hypothesis testing though. *1826

*Number 3, the expected frequency in each category so once you once you compute all the expected *1832

*frequency in order to compute your chi-square, that needs to be each cell each square needs to have an *1840

*expected frequency of five or greater, here is why. *1850

*You need a big enough sample, if you have to small of the sample, again expected frequencies less than five *1854

*also unique big enough proportions, so let us say you want to compare proportions that are like you know *1862

*like one candidate is going to be predicted to win 99.999% of the votes and the other candidate is only *1871

*supposed to win .001% of the vote and you only have five people in your sample. *1883

*And so you need to also have big enough proportion and these balance each other out. *1890

*If you have a large and a sample than your proportions can be smaller also, if you have large enough *1897

*proportions in your sample could be smaller. *1903

*And the final condition is not really condition it is just sort of something I wanted you to know at the rule. *1905

*The chi-square goodness of fit test so that is always been talking about so far.*1913

*This test actually applies to more than two categories. *1920

*You do not just have 2 categories, you have 3 or 4 or 5 or 6 but they do need to be mutually exclusive and *1927

*each outcome in the population must be able to fall into any one of those. *1935

*So those are the conditions. *1940

*So now let us move on to some examples. *1943

*So the first example is the problem that we already looked at so far according to early polls candidate A *1947

*was supposed to win 63% of the vote and B was supposed to win 37%. *1953

*When the votes are counted, A won 340 votes while B won 166 votes. *1958

*One of the things that I like to do just to help myself is when I think of the null hypothesis, when I think of*1967

*the null hypothesis, I sort of write it in a sentence that the proportion of votes, that is my population, *1975

*should be like predicted proportions, and the alternative is that at least one of the proportion of votes will not be like predicted population. *1990

*What I also like to do is I like to draw this out for myself, I like to draw out the predicted population so I will*2032

*color candidate A in blue so that will be about 63%, candidate B will be in red, 37%. *2040

*And so eventually I want to know whether this is reflected in my actual votes. *2053

*The significance level we can set it up .05 just set of convention and we know that it has to be one tailed *2059

*because this is definitely going to be a chi-square and we know it is a chi-square because it is about expected proportions. *2068

*So now let us set our decision stage. *2075

*Now our decision stage, it is helpful to draw that chi-square distribution and to sort of label it, for alpha*2081

*here this is our rejection region .05, now it would be nice to know what our critical chi-square is, and in *2100

*order to find that we need degrees of freedom and degrees of freedom is the number of categories, in this *2111

*case 2 -1 and that is 1° of freedom and it is because if you know let us say that candidate B won that is *2119

*supposed to win 37% of the votes you could actually figure out candidate A like you do not need me to tell *2131

*you what that is to figure it out and candidate A cannot vary, the proportion cannot very freely once you *2138

*know this one and that is why it is number of categories – 1.*2143

*So now that we have that you might be useful to look at either in the back of your book or use XL *2148

*spreadsheet Excel function in order to find our critical chi-square. *2156

*So in order to find chi-square there are two functions that you need to know just like T this and T, F this and F in, now there is chi-this. *2161

*Actually we need to use chi in right now because here we have the probability .05 and the degrees of *2182

*freedom one and that will give us our critical chi-square and that is 3.84. *2190

*So critical and so this is the boundary were looking for 3.84 so anything more extreme more positive than *2198

*3.84 and were going to reject our null hypothesis. *2208

*So now that our decision stage is set, now it is helpful to actually work with our population and remember *2214

*when we talk about our population, should have left myself some room, when we talk about our actual sample here is what we ended having. *2221

*We have observed frequencies already so for candidate A, I am going to write a column for observed in *2236

*candidate B so candidate A, we observed 340 votes so that is our observed frequency for candidate B, we see 166 votes. *2243

*Now one that helps is we know what the total number of votes was, so the total number of votes is going to be 340+166 and that is 506. *2261

*So 506 people actually voted in this so down here I am going to write total 506. *2274

*Now the question is what should our active frequencies have been? *2283

*So here I am going to write expected and I know that my proportion of expected should be 63%. *2291

*That means is that the total number of people who voted? *2298

*So here is our little sample of 506 people. *2302

*This is our 100% but here we have 506 people in our sample, we should expect 63% of 506 to have voted *2308

*for A, and so how do we find that? *2323

*Well we are going to multiply 63% to 506 to find out how many votes that little blue bit is and so that is *2328

*going to be.63×506 that total amount. *2341

*If we multiply 506 x 1 we would get 506 right?*2350

*So if we multiply by a little bit of a smaller proportion that we get just that chunk. 318.78 actually I am *2355

*going to put this here, let me actually draw this little table right in here because that can help us do our 3939.1 finder chi-square much more quickly. *2367

*And so observed expected frequency observed frequency at 340 and 166, okay. *2383

*So what are the other expected frequency for B, so in order to find this little bit we are going to multiply*2394

*.37×506, so .37x506 and that is 187.22. *2401

*And usually if you add this entire column that you should get roughly a similar total. *2414

*When you do it, when you do these by hand sometimes you might not get exactly the same number it *2422

*might be off by just a little bit because of a rounding error, if you round to the nearest 10th, round to the nearest integer, *2429

*you make it a little bit around it here but you should be off by much so that one way you could check to see what you did was right. *2438

*And so once we have this, so let me just copy these down right here so 318.78 and 187.22 for each of these *2445

*the total is 506, so here, one of things we see is that the expected value for A are a little bit lower and the*2463

*expected values for B are little bit higher, but is this difference in proportion is that significant is that *2476

*standing out enough, and in order to find that we need to find the chi-square, the sample chi-square. *2485

*Now, we completely run out of room here. *2493

*But I will just write the chi-square formula up here. *2497

*So the chi-square is going to be the sum over all the categories of the observed frequency minus the *2500

*expected square as a proportion of the expected frequency. *2510

*And so what I am going to do is calculate this for each category, A and B and then add them up.*2517

*So right here I am going to call this a column, O minus E squared all over B. *2525

*So I am going to do that for A and B and then sum them up. *2540

*So, my observed minus expected squared all divided by expected and so here I get this proportion and I am *2547

*just going to copy and paste that down here and then here I am just going to some them up and I get 3.817. *2565

*We are really close but no cigar so where were right underneath so our sample chi-square is just a smidge *2577

*smaller than our critical chi-square so here were not rejecting the null, we are going to fail to reject the*2589

*null, so let us find the P value so in order to find the P value you could use chi disc or alternatively look it up*2597

*in the back of your book, look for the chi-square distribution. *2609

*It should be behind your normal, your T, your F and then chi-square should come right behind it, it usually goes in that order , maybe a slightly different order.*2614

*And our degrees of freedom remain the same one and so all our P value is just over .05, if we round, .51 right? *2627

*So because of that we are not going to reject the null so we are going to say the proportions of votes are roughly similar to the predicted proportions. *2640

*Well, they are not significantly different at least, they are not super similar but we cannot make a decision*2657

*about that but we can say they are not that different from, that they are not extremely different at least. *2663

*Okay, example 2. A study ask college students could tell dog food apart from expensive liver pâté liverwurst and spam. *2669

*All blended to the same consistency chilled and garnished with herbs and a lemon wedge, just to make it pretty. *2684

*Students are asked to identify which was dog food. *2695

*Researchers wanted to test the probability model where the students are randomly guessing. *2698

*How would they cast their hypothesized model? *2703

*Okay so see the download that shows how many students picked that item to be dog food, so it seems that *2707

*college students have a bunch of different choices in dog food liver Patty, liverwurst and spam, and then *2714

*they need to identify which was dog food so out of those, which of those is dog food? *2723

*So it is sort of like a multiple-choice question. *2728

*So if you hit example 2 in the download that listed below, you will see the number of students is selected that particular item as dog food. *2732

*Now be careful because some people right here, remember, you will really get this problem on a test and you would not know that it is a chi-square problem. *2741

*Sometimes people might immediately just think I will find the means and so they just go ahead and find the *2751

*mean but then if you do find the mean, ask yourself, what does this mean? *2758

*What is the idea or the concept? *2763

*If we average this, we would find the average number of students that selected any of these items as dog *2768

*food and that sort of a mean that does not make any sense right? *2775

*And so before you know, go ahead and find the mean, ask yourself whether the mean is actually meaningful. *2779

*So here we know that the chi-square because the students are choosing something and it is a categorical choice. *2788

*They are not giving you an answer like 20 inches or 50° or I got 10 questions correct right? *2798

*They are actually just saying, that one is dog food and they have five different choices and they have *2804

*chosen one of them as dog food so out of five choices of probability model that are just guessing would *2813

*mean that 20% of the time they should pick pâté, once we dog food, 20% of the time don't expand to be *2821

*dog food 20% of the time to pick dog food to be dog food and so on and so forth. *2828

*So let us try that probability model and by model we also need null hypothesis. *2835

*Model or hypothesized population so step one. *2844

*So the null hypothesis is the idea that they will fit into this picture so this is the population, and it is out of*2848

*100% and they have five choices of pictures just lightly un even, it helps really draw this is as well as you can, just as then it will help you reason to. *2858

*That they will have a equal chance of guessing either one of these and there is two liver patties that is why there are 5 choices. *2878

*So liver pâté 1, spam was next, then actual dog food just in the data set, patty 2 and a liverwurst. *2885

*So these are the five choices and were saying look the students are just guessing they should have a 20% probability of each. *2909

*Is this the right proportion for this sample, is the sample going to serve match that or be very different from this. *2923

*The alternative is that at least one of the real proportion is different from predicted. *2938

*So once we have that, we can set our alpha to be .05 our decision stage, could draw there chi-square and *2954

*our degrees of freedom, we now have five categories and so our degrees of freedom is 5-1 which equals 4 *2970

*and it is because once we know four of this, that we could actually figure out the proportion for the fifth one just from knowing 4 of this. *2978

*So that one is no longer free to vary, it does not have freedom anymore. *2987

*So what is our critical chi-square? *2991

*Well, if you want to pull up your Excel data, here I am just in a start off with step three, in step three we are*2998

*critical chi-square in order to find that we can use chi-in, put in the probability that were interested in and our degrees of freedom which is 4. *3011

*And so our critical chi-square is 9.49. *3026

*Noticed that as degrees of freedom goes up, what is happening to the chi distribution is that it is getting *3035

*fatter it is getting more variable and because of that we need a more extreme chi-square value. *3053

*So that is sort of different than like T distributions or F distribution. *3059

*Those distributions got sharper when we increased our degrees of freedom , chi distributions were the opposite way. *3066

*Those district chi distributions are getting more variable as degrees of freedom goes up. *3075

*So once we have this now we could start working on our actual data, our actual samples. *3080

*So step four is we need to find a sample chi-square and in order to do that it helps to draw out that table so*3089

*the table might look something like this. *3102

*I will just copy this down here and this is the type of food, so that is the category and here we have our observed frequencies. *3106

*The actual number of students that pick that thing to be dog food. *3125

*So here we seen one student pick pâté, one to be dog food, 15 students picked liverwurst to be the dog food. *3130

*What are the expected frequencies? *3138

*Well in order to find expected frequencies we know that the expected proportions are going to be .2 all the way down.*3142

*20% 20% 20% 20% and here I am just going to total this up. *3153

*And I see that 34 students were asked this question. *3161

*Are expected frequencies should add up to about 34? *3170

*Are expected proportions adds up to one? *3175

*And that is why we cannot just directly compare these two things, they are not in the same sort of currency *3179

*yet, you sort of have to change this currency into frequency. *3184

*So how do we do that? *3189

*Well we imagine here are all 34 students take 20% of them, how many students will that be? *3192

*So that is 0.2×34, this times 34. *3199

*And I am just going to lockdown that 34 because that total sum would not change. *3207

*So, this is what we should expect that if they were indeed guessing, this is the expected frequencies that *3214

*we should see and if I just move that over here , we will see that that also at the column also add up to 34. *3226

*Now once we have that we can compute our actual chi-square because remember that observed frequency *3233

*minus expected square divided by expected as a proportion of expected. *3240

*So, that is the observed frequency minus expected frequency squared divided by the expected frequency. *3247

*And I could take that down for each row and then add those up and here I get my chi-square statistic for *3257

*my sample and so my sample chi-square is going to be 16.29, and that is the larger more extreme chi-*3268

*square than my critical chi-square, and let's also find P value here. *3281

*In order to find P value I could use chi-disc, here I put in my chi-square and my degrees of freedom which is 4. *3286

*And so that is .003 and that is certainly smaller than .05 and so in step five, we reject the null. *3297

*Now I just want to make a comment here. *3315

*Notice that here, after we do the chi-square although we reject the null just like in the ANOVA we do not *3318

*actually know which of the categories is the one that is really off. *3325

*This one here, we can sort of see, this one probably seems to be the most off but we are just eyeballing it,*3330

*were not using actual statistical principles. *3340

*So once you reject the null there is a post hoc test that you could do but we are not going to cover those here. *3343

*So it seems that students are not randomly guessing they actually have a preference for something as being dog food. *3349

*My guess is liverwurst. *3362

*So example 3 which of these statements describe properties of the chi-square goodness of fit test? *3365

*So if you switch the order of categories the value of the test statistic does not change, that is actually true it*3376

*does not matter whether candidate A got added before candidate B addition is totally order insensitive you *3383

*could add A or B or B on A, you can add pâté or liverwurst and dog food or dog food the liverwurst and *3391

*pate, it does not really matter so this is actually true, as a true property. *3398

*Observed frequencies are always whole members that is also actually true because when you observe of *3403

*the frequency, you are actually counting how many category numbers you have so counting is going to be made up of whole numbers. *3410

*Expected frequencies are always whole numbers, that is actually not true, expected frequencies are predicted frequencies. *3418

*It is not that at any one time you will have plenty student saying that liverwurst is dog food but it is that on*3427

*average that is what you would predict given a certain proportion and so this is actually not true, expected*3435

*frequencies do not have to be whole numbers because they are theoretical, they are not actually things that we counted up in real life. *3445

*A high value of chi-square indicates high level of agreement between observed frequencies and the expected frequencies. *3452

*Actually if you think about the chi-square statistic, this is actually the opposite of what is the real case. *3462

*If we had a high level of agreement this number would be very small and because this numerator is small *3472

*the chi-square would also be small, a high value of chi-square would actually mean that this is quite large *3479

*compared to this and so this is actually also wrong, the opposite. *3486

*So that is it for chi-square goodness of fit test, join us next time on educator.com for chi-square test of homogeneity.*3494

*Hi, welcome to educator.com. *0002

*We are going to talk about the chi-square test of homogeneity. *0002

*Previously we talked about the chi-square goodness of fit test now were in a contrast that with this new test is still 0018.3 chi-square test but it is a test of homogeneity now. *0005

*We are going to try and figure out when do we use which test. *0022

*The test we are testing a new idea , we are not testing goodness of that would actually testing homogeneity similar. *0027

*We actually have slightly different null hypotheses and alternative null and alternative hypotheses . *0035

*We are going to talk about how those have changed then we are going to go over the chi-square statistic and also finding 0051.0 the expected values is going to be a little bit different in test of homogeneity . *0041

*Finally working to go through chi-square distributions as well as degrees of freedom and the conditions for the test of homogeneity, *0055

*one can you actually care conduct this test service statistically legally. *0065

*Okay so the first thing is what is the difference between the test of homogeneity and test of goodness of fit? *0069

*Well in the goodness of fit hypothesis testing we wanted to determine whether sample proportions are very different from hypothesized *0082

*population proportion one way you could think about this is that you have one sample and you are comparing it to some hypothetical population. *0089

*In test of homogeneity and I called it goodness of fit, it is about how well these two things fit together. *0098

*How well does the sample fit with the hypothesized proportion. *0108

*In test of homogeneity homogeneous means similar right, that they are made up of the same stuff. *0112

*In test of homogeneity we want to determine whether 2 populations that are sorted into categories share the same proportions or not. *0120

*And here you could also substitute this word population here because ultimately were using the sample as a proxy for the population. *0130

*So here we have 2 population and we want to know whether those two populations are similar in their proportions or not *0142

*right were not comparing them to some hypothesized population were comparing them to each other. *0152

*And so really you can think of this as an analogy you think of the their relationship by using an analogy from the *0159

*one sample to the independent samples t-test. *0167

*In the one sample t-test we had one sample and we compared it to the null hypothesis right? *0170

*That was when we would have null hypotheses such as new equals zero or new equals 200 or new equals -5 versus an independent sample. *0176

*We had 2 samples and we wanted to know how similar they were to each other right or how different *0190

*they were from each other and our null hypothesis was changed to something like use of X bar minus Y bar equals zero right, *0198

*that they are either made up of the same mean or different means. *0208

*And in a in a similar way the goodness of fit chi-square is really asking whether this proportion in my sample *0213

*is similar to the proportion in our population. *0229

*So that is how I am comparing , this is my null hypothesis in some ways . *0232

*In our inner test of homogeneity we have 2 sample 2 population 2 sample that come from 2 unknown population and we want to know *0240

*whether these have similar proportions to each other and so that is going to be our null hypothesis that these have the same proportion or have different one. *0255

*For null hypotheses is similar proportion. *0267

*And so in that way I hope you could see that goodness of fit in homogeneity their ideas that we have looked at before *0275

*comparing one sample to a hypothesized population or comparing two samples to each other but we have looked at it before *0285

*not with proportion but with means, right? *0294

*And now are looking at it with proportion okay since you are looking at proportion we should have hypotheses about *0297

*proportion so the null hypotheses with something like this the proportion of all the each category the proportion that *0305

*all into each category is the same for each population so however many categories you have so let us say we have *0313

*in a three categories. *0322

*If we believe that they are the same and they should roughly have the same proportion so these have similar proportion. *0341

*It does not actually matter what the proportions are it could be 90, 10 could be 10,10 it could be 75 20 like when the proportions *0347

*that were think there similar for each population and whatever 780 whatever category is 75% of the population *0360

*that category will also be 75% of the population. *0368

*The alternative hypothesis says that for at least one category the populations do not have the same proportion so just like before *0371

*were now talking about differences that the differences are really in the proportions the predicted the populations proportion. *0383

*So just to give you an example. *0394

*Here is the problem and let us try to change it into the null hypothesis as well as alternative hypothesis. *0396

*So according to a poll for and six Democrats said they were very satisfied with candidate A while 510 were unsatisfied *0401

*however 910 Republicans were satisfied with candidate a while 60 were not. *0410

*And in a chi-square test of homogeneity we could see whether the proportions of Democrats and Republicans that Democrats were satisfied are *0415

*similar to the proportions were Republican of Republicans were satisfied versus unsatisfied. *0427

*So let us draw this out first. *0436

*So here we have about 400 Democrats saying there satisfied while 500 saying unsatisfied. *0439

*Let put satisfied in blue and so that is a little bit less than half and the unsatisfied people are a little bit *0451

*more than half so this is the Democratic population that they look like. *0460

*The Republican population looks very different so here we see most of the Republicans being pretty satisfied and *0467

*only a very small minority being unsatisfied right. *0479

*And so the question is are these two are the two similar are the proportions that fall into each category *0483

*satisfied or unsatisfied the same for each population? *0493

*Are they different? *0497

*The null hypothesis would probably say something like this. *0498

*The proportion of satisfied and unsatisfied people like us are similar are the same for Dans as well as republicans. *0501

*The alternative hypothesis says for at least one category either satisfied or unsatisfied, Dans and Republicans do not have the same proportion. *0531

*Okay so note that in the case of 2, once category changes once the proportion of one category changes the other one automatically changes.*0561

*So if we somehow were able to change has satisfied the Democrats were with candidate A, we would also see the *0584

*proportion of unsatisfied people just automatically change. *0592

*So that is in the case of two categories but in the case of multiple categories maybe 2 might change but the others may *0595

*not change right so in that way this would be a more general way of saying alternative hypothesis. *0606

*Now let us talk about the chi-square statistic. *0612

*Now the nice thing about the chi-square statistic is that it is the same as the goodness of fit test. *0616

*We use the same idea so chi-square is going to be observed frequencies and the difference between that and *0621

*expected frequencies where over the proportion of expected frequency. *0631

*But there is just one subtle difference before it was for each category. *0638

*Now we have different categories in different population right so we not only have like category 1 and category 2 *0643

*category 3 so on and so forth but we also have population 1 and population 2 at least right? *0651

*And so we have multiple of observed frequencies and so what do we do right? *0659

*Well what we do here is that we consider each of these combination of which population your in and which category *0668

*are talking about each of these are going to be called cells. *0681

*And so we do this for each cell so I will go from one of to the number of cells. *0686

*And how do we get the number of cells? *0694

*Well the number of cells is really how many population right and that is usually shown in columns times how many categories. *0701

*And that is usually shown in rows, you can also think of the number of cells as columns times rows, how many columns you have times the number of rows. *0718

*But really the idea comes from how many different populations your comparing of chi-square test of homogeneity *0733

*actually compare three or four population not just 2 and how many categories you are comparing. *0739

*So in order to use the chi-square formula, it is often helpful to set up your data in a particular way often *0747

*though that often these formulas will refer to rows and columns and so you really need to have the right data in *0758

*the rows and the right data columns in order for any of these formulas to be used correctly. *0764

*So how to set up your data in this way? *0769

*Whatever your sample one is you want to put that all of the information for sample one into a column, right so *0772

*here I put sample 1 at the generic sample one it could be college freshmen are Democrats or mice got a certain *0780

*drive whatever it is the sample one and these are the people in sample 1 who fell into category one. *0788

*These are the people in sample 1 who fell in to category two and these are called cells. *0798

*When you add these frequency that you should get the total number of people in sample 1 right so in that way all *0804

*the information from 1 one is in a column. *0814

*Same thing with sample 2 all the information from sample 2 should be in a column. *0818

*This should be the entire sample broken up into those that fell into category 1 versus category two and then the*0823

*total gives you the total number of cases in sample 2. *0830

*If you had sample three and four they would follow that same pattern and all the information should be in one column. *0836

*On the flip side when you look at rows you should be able to count of how many people how many cases were in category one. *0843

*And so if you count them up this way this is a sample but it is just how many cases in the entire data set that you are looking at*0855

*are in category 1 and if you look across here this is how many cases in the entire data set fall into category 2 *0868

*and finally if you look at this total of totals what you should get is that is the entire data set all added up. *0878

*So let us try that here with the Democrats and Republican example. *0889

*So I am going to put Democrats appear Republicans appear satisfied and unsatisfied and all I need to do is make *0896

*sure I find the correct information and put it into the correct cells. *0910

*910 are satisfied 60 are not. *0916

*When I add this up I should be able to get the number of how many Democrats total that are in the sample so this *0921

*is 916 for Republicans this is 970 so we have slightly more people in a Republican sample than our Democrat sample and that is fine. *0929

*If I add the rows up like this if I get the row totals what I should get is just a number of satisfied people. *0940

*It does not matter whether their Democrats or Republicans so we should get 13, 16 and this should be 570. *0948

*And if I add these two accession equal these 2 add being added outbreak of interest adding these four numbers up *0959

*in a different order so that should be 1886. *0967

*So we have 1886 in our total data set across both sample and we know how many people were satisfied , how many *0973

*people are unsatisfied we also know how many Democrats we had how many Republicans we have and all the different combination right? *0990

*Democrats are satisfied Democrats unsatisfied Republican satisfied Republicans unsatisfied. *0998

*So this is a great way to set up your data that really can help you figure out expected frequency which is a *1003

*little bit more complicated to figure out intensive homogeneity. *1009

*Not too much complicated but just a little bit more. *1012

*So here is how we can figure out expected frequency so once you have it set up in this way Democrats Republicans *1017

*satisfied unsatisfied, once you have it set up in this way here is the formula used for expected frequency. *1026

*So E is going to equal basically the proportion of people who are in one particular category. *1033

*So I just want to know how people tend to be satisfied. *1042

*I do not care whether their across a Republican, just in general who satisfied right so that would be the row *1046

*total right so the row total over the grand total this one right here. *1053

*This will give me the rates or the proportion of just the general rate of who satisfied who tends to be satisfied *1065

*that 70% to be satisfied 20% to be satisfied 95% to be satisfied. *1077

*What is the general rate and I am going to multiply that by the total number of the sample that I am interested in *1084

*so maybe I am interested in the Democratic sample so I would get the column totals. *1092

*So that is the general formula that will show you this in a more specific way so let us talk about the expected value of *1097

*Democrats who are satisfied. *1107

*Right so that would be the satisfied total over the grand total so this gives us the rates of being satisfied just *1110

*in general what proportion of the entire data set is satisfied and then I am going to multiply that by however *1125

*many Democrats I have so Democrat total. *1132

*So I could write it in this way but what ends up is that this is just a more general way of saying this example. *1137

*So when I say Democrats total is the same thing as being column totals. *1146

*And when I say row total it is really the same thing as being satisfied total and the grand total is the total number in our data set. *1151

*Democrats Republicans. *1162

*So now let us talk about once you have the expected values you have the observed frequencies and now you could easily find chi-square. *1165

*Once you get your chi-square how do you compare it to the chi-square distribution? *1176

*Well the nice thing is the chi-square distribution looks the same as in the test at as in the goodness of fit test *1182

*and so chi-square it has a wall at zero can not be lower than zero and it has a long positive tail and when you decide how much *1190

*your alpha is and that is what it is going to look like Alpha is always one tailed in a chi-square distribution. *1202

*But the question is how to find degrees of freedom now that we have rows and columns? *1208

*Well the degrees of freedom is really going to be the degrees of freedom for category times the degrees of freedom for *1217

*however many populations or sample that represent your population you have and that is going to be the number of rows *1229

*right because each categories in a row -1 times the number of columns you have -1 so that is how you find you degrees of freedom *1238

*when you have more than one population that you are comparing. *1248

*So what are the conditions for the test of homogeneity? *1251

*These conditions are to be very similar to the conditions for out goodness of fit testing so the first thing is *1258

*each outcome of each population falls into exactly one of the fixed number of category. *1265

*Well the categories are mutually exclusive just like before, you have to be in one or the other you cannot be into 2 categories*1275

*at the same time you cannot opt out of being in a category also the category choices must be the same for all population. *1280

*So it went to one population has to have if they have three choices the same three choices must be the case for population 2. *1288

*The 2nd requirement for condition is that you must have independent and random sample before in tests of goodness of fit *1298

*we only have this requirement that the sample have to be branded because we only had one sample. *1310

*Now we have multiple samples and they must be independent of each other they cannot they cannot come from the same pool. *1315

*So third condition the expected frequency in each cell is five or greater and not just is the same condition that we had *1325

*for goodness of fit testing it is because you want a big a sample as well as the big enough proportion. *1337

*And number four is not really a condition is just so that you know how free you are with chi-square testing you can have *1344

*more than two categories and you can have more than two populations you could have 4 categories and six population so you *1355

*should have a whole bunch of these different combination so you are not restricted to 2 categories and 2 population. *1364

*So now let us go on to some examples. *1371

*Example 1 is just the example we have been using to talk about how to find how to set up your data and how to find *1376

*expected values so I set this up in an Excel file this is just exactly the same way we set it up previously I just found *1383

*the row totals as well as the column totals. *1397

*And now I could start of my hypothesis testing so first things first. *1400

*Step one our null hypothesis should say something like this that the proportions of satisfied and unsatisfied people minus adults *1406

*for Democrats should be the same as for Republican so the proportion of category one and two of satisfied and *1425

*unsatisfied by Allstate voters should be similar for Democrat and Republican. *1435

*So the alternative hypothesis is that at least one of those proportion will be different between Democrats and Republicans. *1446

*Step two, just set our alpha to be .05 and we know that because we are doing chi-square hypothesis testing is one *1461

*tailed step three you might want to draw a chi-square distribution for yourself or just in your head and certain *1476

*color and that alpha part and try to think. *1485

*I want to find my critical chi-square. *1488

*In order to find the critical chi-square I need to find the degrees of freedom. *1493

*And my degrees of freedom is going to be made up of the degrees of freedom for categories as well as the degree of *1499

*nfreedom for population and there is two populations so it is 2-1 and you could also see that as the columns 2 column – 1. *1509

*And the degrees of freedom for number of categories is with two categories that is satisfied and unsatisfied -1 *1521

*and there that corresponds perfectly to number of rows -1 and so the degrees of freedom here is going to be that *1535

*this times this so degrees of freedom for category times degrees of freedom for population and is just one. *1545

*So, what is our critical chi-square, but that is going to be found by chi in we put in our probability as well as *1553

*our degrees of freedom and we find 3.84 is our chi-square critical chi-square. *1564

*So we are looking for sample that represent population sample chi-square is that are larger than 3.84. *1571

*Step four look something like this so in order to find your sample chi-square what we need to do first is find our *1584

*expected values so here we have observed frequency and what we need to do is find infected frequency. *1595

*So I am just going to copy and paste this down here so we do not have to keep scrolling and so I am going to draw *1609

*a director at the table here for observed frequency and create the same table for expected frequency. *1623

*Okay so when I look at my expected frequency I need to find out what is the general rate and then multiply it by *1635

*however many however many industry people have in that sample so the general rate of being satisfied is 1316÷1886 *1651

*so that the general rate and that is about 70%. *1670

*Take that and multiply that by the total number of Democrats. *1674

*Now this part I want to keep that the same and I want to keep that in the same column so I am going to put $ affinity *1680

*to walk down that column and here I am going to put $ in front of both the D and the 21 in order to lock down this actual cell. *1697

*Because here is what I am going to do I am than actually copy and paste that over here and if look at this then what I am doing*1708

*is I have this same rates again the rate of being satisfied but now it is multiplied by the number of total Republicans. *1716

*And I am going to take that cell copy and paste it down here and here I see that now I have the rates of being *1726

*unsatisfied and they need to change this to that and here I have the rates of being unsatisfied and then *1737

*multiplied by total number of Republican so these are my expected frequencies. *1750

*Notice that the total still add up to be the same right and usually it should there might be some slight discrepancies*1756

*but that will just be because of rounding error so they should still be pretty close. *1766

*So now we have observed frequencies as well as expected frequencies and now we need to figure out my chi-square. *1771

*My chi-square is going to be made up of observed frequency minus expected frequency squared divided by expected frequency. *1779

*And I am going to need to find that for Democrat Republican as well as satisfied and unsatisfied and then add off all of these cells. *1790

*So I will see grand total and I will put that over here. *1808

*Okay so let us find the observed frequency minus the expected frequency squared divided by expected frequency. *1813

*And I could just copy and paste that here because Excel will just move everything down and I can take this over here because Excel *1829

*will move everything over to the right. *1841

*And the grand total for all four of these is going to be 547.18 and so my sample chi-square is quite large.*1843

*And so do I reject my no hypothesis? *1876

*Indeed I do and we can find the P value so here I will put chi disc in order to find my probability. *1881

*Here it is, degrees of freedom is going to be one and that is a very very very small P value so that is the pretty radically*1898

*different population that we set in there. *1911

*And if you want to step five, example 2. *1917

*Consider this data on pesticide residue on domestic and imported fruits. *1933

*Does this data fit the conditions of a chi-square test of homogeneity regardless of your answer conduct hypothesis tests. *1937

*Now be careful here although you see column and rows these are not the columns and rows you should be using the columns are *1944

*actually okay domestic roads imported roads we could consider those two to be the different populations that are interested in. *1956

*But the roads actually do not show the different categories such as sample size percentage showing no residue and percentage showing residue in violation right? *1964

*So what we should do is we should actually transform this data into sort of the correct setup. *1975

*So here you could just pull up a brand-new XL file just been a user of the bottom portion here and here is what we want. *1983

*We would like it to be set up so that we have the two populations appear and we have the different categories here *2005

*so the categories are probably going to be showing no residue showing residue in violation but one of the things I *2028

*noticed is that these percentages do not add up to 100 that there must be some other category that were missing. *2035

*So no residue showing residue in violation of the law so I guess that is really bad and maybe there is just one *2042

*word it is residue but not in violation and you sort of have to figure that out from the data that they have given you. *2054

*But they do give you the sample size 344 as well as 1136 so this is the total. *2063

*The question is what are our observed value? *2073

*In order to find observed value all we have to do is multiply but the proportion so 44.2% times the total. *2079

*Here I walk down that row, now residue in violation what I have to do is to change this percentage so the percentage is .9%.*2098

*So that is .009 so that is .9%. *2116

*And so what sort of leftover? *2127

*Well, the leftover percentages is 1-.442 + .009 right so that sort of everybody else and that is I guess the *2131

*number of fruits that are not in violation but still have some residue on them, some pesticide residue times this. *2143

*And so when I add them all up I could check and that is 344 so I have done my proportions correctly. *2154

*Now right away we could see that were actually not meeting the conditions for chi-square. *2169

*If you look at this cell right here that has that only has three fruits in it even if we round up generously it is 3.1 right? *2176

*So there is only three fruits. *2188

*Remember expected frequencies have to have at least 5, so here the observed value is pretty small. *2191

*Okay so that it said go ahead into hypothesis testing anyway you should not do this in real life but *2200

*for the purpose of this exercise let us do it. *2210

*So now let us find the proportion of imported fruits that are observed to have no residue on them. *2212

*So that 70% 70.4% times this total and that is almost 800 fruits. *2222

*Also we have those that have residue in violation .036 that is 3.6% times 1136, about 41 fruits and then *2232

*I need the leftover percentage , so that is 1-.70% 74.4% +3.6% . *2249

*That percentage times the total. *2262

*And that is 295 right? *2268

*So first notice that these seem like there is way more of these imported fruit than domestic fruits but that is because the *2272

*totals are different so it does not necessarily mean that imported fruits they have so much residue on them, *2280

*that is not necessarily what it means, but that is hard to compare because they have totally different totals. *2289

*So it is helpful to find the row totals as well because that can help us find expected value expected frequency *2299

*and so that is adding these rows together and we have a total of 1480 fruits Domestic and imported altogether. *2308

*Once we have that then it would be easy for us to find expected frequency and expected frequency we could basically set up in a very similar way. *2329

*So what is our expected frequency? *2346

*Well,expected frequency is generally how frequent with the proportion of no residue over all the fruits right. *2362

*So that will be this row totals divided by the grand total that is the general rates and we want to lockdown this row *2370

*because we want to lock those two values down because and that is always going to be the rate for no residue *2383

*times the actual number of domestic fruits. *2401

*So we get 221 and here we do the same thing and I just copied and pasted across an Excel will just naturally you figure out what to do. *2410

*So this is the rate of no residue over total fruits times the total number of imported fruits. *2428

*Then we find there the rates of fruits that have residue but are not in violation which is this total over the grand total. *2436

*And then I am going to lockdown those values and then I am going to multiply that by the total number of domestic fruit. *2449

*And then if I copy that over that should give me the total number of imported fruits expected value of imported fruits given this proportion. *2467

*And finally the proportion of fruits with residue in violation so a lot of pesticide residue that would be this total *2476

*divided by the grand total times the total. *2489

*And here what we can see is if we sum these three expected frequency together we should get something similar to 344. *2502

*And indeed we do and here we should be 1136 and indeed we do great. *2515

*So once we have our table of observed frequencies as well as expected frequencies now we can start to calculate *2522

*for each cell the observed frequency minus expected frequencies where as a proportion of expected frequency. *2530

*So O minus E squared as a proportion of expected frequency so I will copy this cell labels so observed frequency *2540

*minus expected frequency squared divided by expected frequency , and just copy and paste all that let us check one of this. *2558

*This one says that observed frequency minus expected frequency squared over expected frequency. *2573

*And when we add all of these up we get 102 but we have forgotten the difference as we forgot to make a decision stage*2581

*so let us go ahead and do step three. *2599

*So the decision stage will be our critical chi-square and our critical chi-square sound with degrees of freedom *2601

*of the categories times the degrees of freedom of the population multiplied together so the other degrees of freedom for the chi-square. *2610

*So categories -1 is 2, population -1 is 1, so the degrees of freedom is just 2, so our critical chi-square is chi in. *2628

*Put in .05 as our desired probability, our degrees of freedom equals 2 and we get 5.99. *2646

*We see that our chi-square is much larger than that so we would reject our null.*2653

*Hi, welcome to educator.com. *0000

*Today we are going to overview all the statistical tests we covered so far. *0002

*So this is the last lesson in this series.*0008

*We are first going to list all the statistical tests that we covered. *0011

*In particular we are going to cover the hypothesis test. *0016

*We are going to organized them into a chart so that you can tell which test was performed by looking at a set of results.*0020

*So here is a giant list of hypothesis test that we covered so far.*0031

*Other one sample z-test, the one sample t-test, independent samples T paired samples T one-way ANOVA *0036

*also called the independent samples ANOVA, repeated measures ANOVA chi-square goodness of fit chi-squared test of homogeneity.*0044

*In more advance statistics courses, you may undercover also cover hypothesis testing with regression. *0055

*It does exist however we have not covered it in the set of lesson. *0062

*So the question is how do we know which of these tests that we should perform when we see a set of data *0069

*or how you look at a set of results and figure out which is the test that they did in order to come up to this result. *0076

*It actually helped to organize all of this different type in this table right here so there is a couple of dimension. *0084

*One dimension is how many samples you have, so one sample test, 2 sample tests and more than two sample test. *0093

*Now these hypothesis tests are all similar and that they all require at least one sample and because of that *0102

*they might also be called having a categorical independent variable so that is what they all have in common.*0113

*But they have different levels of the independent variable. *0121

*So this only has one level that has two levels and this has more than two levels. *0125

*But also we need to know what is the measurement what is the dependent variable that they are interested in. *0132

*There might be categorical dependent variables such as are they satisfied or unsatisfied. *0139

*Did they pick red blue or green or there might be continuous dependent variables. *0145

*How much did they improve on a test how fast were they going how many inches did they grow? *0153

*Different DVs like that had a numerical value were we can find the mean as well as the variance and standard deviation. *0163

*When we have categorical DVs such as yes and no where red blue and green we cannot find the meaning of those kind of value. *0172

*So let us start organizing our test. *0183

*When we think about one sample test there a couple of one sample tests we have talked about already. *0186

*Some of them literary have the word one sample in their title such as the one sample Z test and the one simple t-test. *0191

*The one sample Z -test and one simple t-test obviously use the mean as well as standard error which is *0199

*calculated by tabulating standard deviation of the sample so that would fall into the continuous dependent variable box right here.*0207

*So there is the one sample Z as well as the one sample T. *0218

*How you know when to perform the one sample z-test versus the one simple t-test well you know how to do that if you know Sigma. *0227

*So if sigma is known the actual population standard deviation then you go ahead and use the one sample Z- test. *0238

*If sigma is unknown a.k.a. you have to use S instead then use the one sample t-test and that is because the *0247

*T is more variable and it is much more like the normal distribution as N your sample size becomes greater and greater. *0267

*How about the categorical DV which is the one sample tests that we could put in here, well the categorical *0278

*DV that we have looked at are all called chi-squared test. *0288

*So there is a chi-squared test which might be written as chi-square or chi-squared there is a chi-squared test *0292

*that only uses one sample and compares it to a population but here they take that one sample and look at *0305

*the samples proportion and see if that matches the population’s proportion. *0313

*That test is called the goodness of fit test because that goodness of fit is looking at how the sample fit with the population, goodness of fit. *0319

*So, we have already tick-tuck three tests. *0330

*Now let us talk about two sample test, when there is two examples and we often want to look at whether *0338

*those samples are similar in that, the new of one minus the new other equals zero or we want to look at *0349

*whether they are different in that the means of these populations do not equal each other . *0358

*Those tests are called t-test. *0365

*Right so the two sample t-test and obviously t-tests require calculating a T which requires mean standard error standard deviation so does t-test belong in here. *0370

*So the first t-test we learned about where the independent samples t-test, as well as the paired samples t-test. *0384

*This is both t-tests that take into account 2 sample and they have a continuous dependent variable. *0402

*How do we know which one to use well you has to check for whether the samples are actually independent? *0414

*If the samples are independent use the independent samples t-test sort of a no-brainer. *0421

*If the samples are linked in some way then use the paired samples so with independent samples use the *0426

*independent samples t-test with link samples use the paired samples t-test. *0435

*Linked or dependent samples. *0442

*Now what about when you have a categorical DV and you have more than one sample you can no longer *0446

*use the chi-square goodness of fit test instead you have to use the chi-square test of homogeneity . *0453

*This test whether 2 population are similar to each other in terms of their proportion or not just like the t-test*0461

*look at whether 2 sample are similar to each other in terms of their means or not and so in that way these tests all have that in common. *0480

*What they have different from each other that's different from each other is that this chi-square use categorical DV and the t-test use continuous DV. *0491

*So what about if we have more than one sample. *0501

*Well actually if we had more than one sample and we have a categorical DV we can continue to use the chi*0504

*square test of homogeneity because here we can use it for two sample 3 sample whatever however many *0510

*samples you like as long as it is not one so we could just say chi-square tests of homogeneity, and life is simple. *0517

*However if you have a continuous DV now you can use t-test anymore because T-test only compare *0530

*two distribution now we need to compare multiple distribution how do we do that. *0538

*We use the F test also called ANOVA analysis of variance. *0544

*So there are two kinds of analysis of variance test that you learned. *0549

*One was the independent samples ANOVA and the other was the repeated measures ANOVA. *0553

*How do you know which one to use, well it is just like this separation right here with independent samples *0568

*use the independent samples ANOVA with link samples or dependent samples you use the repeated measures ANOVA. *0581

*So that is how we know which test to do so we could look at a set of data look at whether it had *0589

*continuous DV or not look at whether has to samples one sample more than one sample and we could *0597

*follow this chart to figure out which tests should be performed and which does we can perform. *0603

*So now let us practice. *0609

*The following data are from OkCupid, an Internet dating website that does a lot of cool things with the data. *0613

*So you could check out the blog at blog.okcupid.com and many of these figures are adapted from that website. *0621

*The following data may be offensive to some of you because some of the data to mention sex and some of the data mention cleavage. *0630

*Example 1 so here is a statistical conclusion and we need to figure out what statistical tests we should do. *0637

*The statistical conclusion is this. *0647

*The weird MySpace angle profile photo the one it looks like this, that results in more messages than other *0651

*photo contacts, so here are the different photo contacts, things like my space shot in bed, outdoors travel *0659

*with friends and the dependent variable is the new contacts monthly. *0666

*How many new contacts they have per month so these are my two variables, photo contacts as well as number of contacts monthly. *0672

*My number of contacts is my dependent variable and my photo contacts this happens to be my multiple groups, my different samples right. *0688

*So I have a sample of people who has this as their profile shot this is their profile shot is that their profile shot. *0704

*So these are my sample here and I have eight samples with continuous DV so which statistical tests should be performed? *0711

*Well it should be an independent samples ANOVA because we have more than two group, 2 groups and are devious continuous. *0722

*So we can analyze the variance between the groups over as a ratio of the variance within the groups. *0734

*So example 2, use the statistical conclusion straight and bisexual men are more likely to believe they are geniuses than gay man. *0747

*What are the variables and which statistical tests should be performed? *0760

*So they are comparing three different groups of men bisexual men gay men and straight man so that things *0764

*like samples already and what they are asking them is just yes or no. *0773

*Do you think you are a genius are you a genius , yes or no, that is a categorical variable and the we have a *0778

*categorical dependent variable so what statistical tests should be performed? *0785

*Well, three groups in a categorical dependent variable this seems like this seems to call for the chi-square test of homogeneity. *0792

*We want to know whether these three different samples have similar proportion or different proportion. *0802

*Example 3 the statistical conclusion says this. *0813

*Both male and female iPhone users are more promiscuous than blackberry and android users. *0823

*So what are the variables and which statistical tests should be performed? *0829

*This is actually a little bit of a trick question. *0834

*You can answer the best of your ability but I'll show you how to go one step beyond what we actually know, okay. *0837

*So one thing we could do is just compare these three groups of three groups of cell phone users so that *0844

*seems like three samples to me that are independent. *0852

*Usually people do not have more than one cell phone and this looks like the average number of sexual *0855

*partners at age 30 so this is the bar graph right here not a histogram which should be a frequency *0862

*distribution and this seems like a continuous dependent variable. *0868

*After all in order to compete an average you have to have a continuous variable so we have a continuous *0875

*DV with three groups of cell phone users. *0882

*The one answer that we could come up with is to say perhaps the one one-way ANOVA also called *0885

*independent samples ANOVA but and that would be a good answer given what we have learned so far. *0895

*Hopefully you will have learned enough about statistics that you can take multivariate statistics which is sort of the next level . *0909

*In the next level which you will learn about in when you have more than more than two independent variables. *0915

*Here we have independent variable of cell phone as well as the independent variable of gender and when you cross them together we get six groups. *0923

*Android users were male android users who are female and blackberry users or male and blackberry users who are female, iPhone male, iPhone female. *0934

*With six different groups now later on when you look at this factorial ANOVA, they can actually almost like doing 2 ANOVA at the same time.*0946

*And so this would actually technically be a factorial ANOVA but if you can answer the nova you are pretty close. *0958

*So example 4, older women cleavage pictures are associated with greater improvement in monthly contact them for younger women. *0966

*Okay so one of the ways we can look at this is looking at age and we can look at the difference between this *0978

*as the dependent variable and that is definitely continuous and we can look at the difference here as well and compared those 2 differences. *0987

*Just at age 18 and age 32 so we looked at these two groups of women that so the 18-year-old women and the 32-year-old women. *0997

*We look at those two groups of women and look at the DV of improvement how much improvements what kind of test would we do? *1008

*Well it seems as though we should do a t-test of some sort because this is a continuous variable and we *1021

*have 2 groups and the groups seem independent. *1030

*We cannot be 18 and 32 at the same time and I do not think they are following the 18-year-old until they *1033

*become 32 so I do not think they are linked so it seemed like an independent samples t-test. *1041

*But there are other ways you can look at this, you can look at this as a regression correlation you can look *1046

*at the regression line for women with women showing cleavage in light blue and women not showing *1061

*cleavage in the dark, dark blue so you can look at those two regression line so that is another way that you could go on this. *1072

*So that is the end for statistics on educator.com, thank you so much for watching.*1083

*Welcome to www.educator.com.*0000

*Today we are going to talk about samples and about cases, variables, and measurement within samples.*0002

*We need to talk about samples because statistics is all about data and data is made up of cases, right?*0012

*Each individual that is part of that data set is called the case, and cases are actually made up of variables.*0019

*You could think of variables as different characteristics within a case and a variable can take on different values.*0028

*Just to give you an example here is the data set that is simple and we have three cases, 3 shapes and they have different variables.*0037

*You can think of these as dimension.*0048

*Dimensions of shape, color, area, right?*0051

*These variables right up here, these can actually have different values.*0056

*For instance, triangle is the value for this case, for this variable of shape.*0068

*For this case, square is the value for the variable of shape and circle is the value for the variable shape for this case.*0076

*A variable can take on different values and because of that it is called a variable because it could vary.*0091

*It does not have to vary for instance take a look at color right here.*0099

*This is a variable that has all of the same values, teal.*0103

*Although they do not have to, in a variable the values do not have to vary, they can.*0110

*We could put a red case in there and it is okay.*0119

*One thing to note is that regardless of data sets oftentimes you will see cases listed in rows.*0127

*Often each row is the case.*0134

*Also often each column is the variable and you will learn about different kinds of variables as we go on.*0136

*When you look at columns you see variables.*0144

*When you look at entire rows you see cases.*0147

*Not only that but when you look at a cell, a cell is a combination of a particular row and a particular column.*0151

*When you look at a cell, that cell often contains a value.*0159

*The next two cases, variables, and values, as a small note about where they might be in space you might say usually in rows.*0164

*This one is usually in columns and values are usually in cell.*0178

*Does it always have to be the case but usually by convention many data sets are organized like this.*0186

*We can look here.*0194

*Here the cases seem to be made up of individuals.*0195

*Here the individuals are taken from www.facebook.com.*0200

*The variables are things like gender, friends, siblings, and number of tagged photos.*0204

*Tagged photos by itself is the variable, it could vary.*0213

*There are lots of different values that it could hold.*0217

*For instance 24, 42, and 21.*0220

*These are different values that could be sort of sitting in the place of the variable tagged photo.*0224

*Just to give you one more example, here is an example of aircrafts.*0236

*These cases are aircrafts and on each row there is information for this particular aircraft on that row.*0241

*The different variables here are number of seats, the cargo that it can carry in tens, and let us say average flying speed.*0250

*Here we could see that the B747 has 410 seats as the value for the variable number of seats.*0262

*Once again it is organized, rows being cases, columns being variable and cells being values.*0272

*I want to introduce one other idea.*0285

*Remember I said that variables can have different values, they do not have to differ but they can.*0288

*There are some characteristics that will not vary though because of a particular design of the study.*0296

*For instance, maybe a study would like to look at a pregnant women *0301

*and how much prenatal exercise they do and whether that predicts the health of their baby.*0306

*Because of the design of this study, the variable gender is actually not going to be a variable *0316

*because there are very few of them doing prenatal exercises because they are pregnant.*0324

*Instead this is what it is going to be called a constant because the values are all the same by design, are defined.*0329

*The question is great we know how to organize the data once we get it but how do we actually get that data?*0340

*The process of getting new data is called research and often research is taught with the five scientific steps and asking a question, *0348

*coming up with the hypothesis, coming up with the design, research analysis, and coming up with the conclusion.*0359

*That sort of addresses that question.*0366

*In order to reframe the 5 steps of science so that it relates more to statistics *0369

*I’m going to talk about these things in terms of cases because that is what is involved in statistics.*0379

*Research will be about how to get the sample.*0388

*Already we are putting in our statistics terms, how to get the sample.*0400

*The research question is often a proposed relationship among variables.*0404

*A hypothesis often goes with that so it is says yes I do think this is the relationship or I think there is another relationship. *0419

*These often go together.*0430

*The research design is the procedure that we use for actually collecting the data.*0432

*Measurement is actually the process for gathering quantitative information that represents some variable or variables.*0451

*Let us say the quantitative values just to use the same words.*0470

*Values that represents or variables.*0475

*Here we are talking about how to actually get the sample.*0493

*We are looking at proposed relationships among variables within those cases.*0496

*Research design is all about the procedure for collecting that data.*0502

*Measurement is about gathering quantitative values that represents some variables.*0507

*Research analysis is what we often think of when we think of statistical analysis so I put statistics right here.*0514

*Here in statistics, there statistical analysis is going to have its own statistical question and hypothesis.*0522

*It is also going to have statistical procedures.*0533

*You are going to be able to come up with statistical conclusion.*0540

*Often this little mini set is often called hypothesis testing.*0547

*We will get to that when we talk about inferential statistics towards the middle and latter end of the course.*0563

*Finally the research conclusion is going to be different than the statistical conclusion.*0572

*Here in the research conclusion we step out again and go back to how this analysis relates to this overall research question.*0577

*This is the general conclusion.*0588

*This general conclusion is created from the statistical conclusion as well as in considering all that came ahead of you.*0594

*What kinds of variables are there if our research question and our hypotheses are all going to be made up of variables *0607

*we better try to figure out what kind of variables could there be?*0614

*There are a couple of different variables that you need to know.*0619

*When we already covered this one is not a variable it is right outside the border in variable but it is related.*0622

*A constant is the characteristic that cannot vary in the data set.*0630

*For whatever reason it cannot vary but other than that they are two kinds of variables you need to know.*0633

*One is discrete variables and when we talk about discreteness, we are talking about things that have very particular values.*0639

*When you think about a number line there are only certain places that can contribute a value to a discrete variable.*0650

*These are the only values sort of allowed in a discrete variable.*0665

*Example might be something like number of siblings, you may wish you had only one and a half sibling but that is actually not possible.*0670

*Number of siblings is what we think of as a discrete variable.*0680

*You either have 1 or 2, you rarely have 1.65 or 1.82 number of siblings.*0686

*Also another example might be number of gold medals won in the Olympics.*0695

*Often people do not win just half a medal or 1/8 of a medal, or 5 2/6 of the medal.*0706

*Instead they win whole medals.*0715

*There is only particular place on the number line that can contribute values to these variables.*0717

*These are examples of discrete variables.*0725

*Continuous variables are exactly the opposite.*0728

*We might have these in a whole numbers like 1, 2, 3, 4 but when you have a continuous variable *0734

*you could have this be the value or you could have this be the value or one right next to it as the value or over here as the value.*0740

*Any of these values can contribute to the variable.*0748

*One way you might want to think of this is that there are no gaps on the scale.*0753

*Any value can contribute, can be part of this variable.*0763

*In discrete variables only certain values can take part in this variable. *0769

*Examples of continuous variables are things like length, weight, these are values that can have any number.*0777

*It does not have to be 100 or 101, it could be 100.1 or 100.001, or 100.0001.*0794

*There is an infinite even between 0 and 1, there is an infinite number of values that *0810

*could contribute to a continuous variables such as length or weight.*0816

*Other possibilities are more abstract, things like anxiety level or knowledge of history.*0822

*Somebody could be maybe right here in terms of anxiety level but someone else could be very close but just less anxious in them.*0833

*These are what I thought of as continuous variables because any value is actually possible.*0847

*Here is the thing, we cannot actually quite get variables in the world.*0858

*We cannot get the batch true, instead we have to measure it and often measurements are almost all discrete.*0864

*When you actually measure something we often round, for instance when we measure height we do not measure it to the .0001 inch or centimeter, *0873

*instead we often round it to the nearest whole unit.*0885

*Often people do not say I’m 5’6 and 375 of an inch.*0891

*Often people do not say that and because of that most measurements are actual scale of getting values of the variables.*0901

*Those end up turning all variables into discrete variables.*0912

*But underlying the variable, it does not have to be discrete just because we measure it in that way.*0918

*When a variable is measured you will end up with a particular set of numerical values.*0925

*That is often what we think of as our sample distribution, our scatter of numbers.*0930

*It often helps to ask ourselves what kind of scale is it on.*0937

*It is all going to be discrete but there are different levels of in formativeness that measurement scales can give us.*0943

*Let me give you some examples.*0953

*One reason that it might be helpful to think about what kind of measurement scale a piece of data is on is because it helps us compare pieces of data.*0958

*For instance could we look at number of friends and compare that to ranking in class.*0968

*Those numbers actually stand for very different ideas and that is what we mean by measurement scale.*0975

*What does the number mean?*0983

*What kind of information does it give us?*0985

*When we think of something like gender, here we are using a number 1 and 2 *0988

*but are we saying that somehow 2 males if you add them together you get a female?*0994

*Is that what we really think? Not really.*1001

*These numbers are just stand ins for other ideas.*1004

*When we are talking about number of friends, if we had somebody who has 48 friends, *1009

*we do mean they have approximately 1/4 of the friends that the second person has.*1015

*Can we compare ranking in class?*1021

*Is this person somehow too better than this person? How do we compare?*1025

*It often helps to know what kind of measurement scale we are working with.*1034

*There are four different kinds of measurement scales you need to know. *1039

*Here they are nominal, ordinal, interval, and ratio and I have listed them in an order where they become progressively more informative.*1044

*There is more and more information as we sort of go down.*1054

*These are the types of skills you might run into.*1057

*Nominal scales are often referred to as dummy codes because nominal scales are just numbers that stands for names.*1061

*The look on the surface like numbers but they are just names and the numbers do not actually have any meaning.*1071

*There is no meaning in the number, they just stand in like a dummy for a name or category.*1079

*Right so nominal scales stands for the idea name.*1086

*You can think of this is a qualitative scale, there is no order.*1094

*Some examples might be things like color of eyes, there is no order.*1109

*It is not that blue has to go before brown, or green has to come after brown.*1113

*There is no particular order to it.*1117

*Another idea that nominal scale is political affiliation or type of major.*1121

*These are nominal scales because it is not that there is any inherent order.*1129

*Even if we assign numbers to it, the numbers are just arbitrary, they do not actually mean anything.*1132

*Things like types of cheese, state that you come from, what language you speak, those are all examples of nominal measurements.*1140

*The second level we can think of measurement, it has a little bit more information.*1152

*It is no longer just a stand in, here we now have an order.*1160

*The numbers actually tell you about order but they may have uneven intervals.*1166

*1 and 2 are not the same distance apart as 2 and 3.*1174

*A good example of this is Olympic gold medal, silver medal, and bronze medal.*1186

*When we think of gold medal, silver medal, and bronze medal, and let us think of this is how the long jump.*1192

*The gold medal may have jumped this far.*1206

*The silver medal may have been very close.*1210

*But the bronze medal may have been far off.*1213

*But when they actually get their medals you cannot tell how far off each one was.*1217

*You do not know whether the intervals are the same or different.*1223

*Here we preserve order.*1227

*Now when we know the number 1 and 2 we know that number 1 definitely comes before number 2 *1229

*but we do not actually know the interval distance between them.*1235

*Other examples of ordinal scales are things like your rank in law school, *1240

*that ranking number does not actually tell you how much better someone is than someone else.*1246

*They might be very close but their numbers might say they are one apart.*1253

*Often examples of things that are ordinal are often rank ordered.*1261

*Whenever you hear the word rank, that is often our ordinal scales.*1266

*Things like having a Masters degree, PhD or bachelors degree, those have ordinal scales.*1272

*They have order in terms of how much schooling you had to do but they do not necessarily have the same distance between them.*1281

*Now we get to interval scales and remember I said it is more and more informative as we go down, *1296

*now we have order as well but also even intervals.*1300

*The distance between 1 and 2 is the same as the distance between 2 and 3.*1307

*When we have interval scales you might think that is like a regular number of line.*1313

*There is one thing that this scale is missing, although it has order and an even intervals there is no meaningful 0.*1321

*Here is what this means usually when we have a meaningful 0 then that would mean that when we say there is 0 of this, *1331

*then there is literally none of whatever it is.*1342

*In an interval scale it is relative.*1347

*It does not matter whether you start marking out 1 or whether you start marking at 0, or whether you start marking at 125. *1350

*Let me give you an example that is commonly used especially in the social sciences.*1359

*Often when people are asked about their opinion in self report, they are asked to rate something.*1363

*How happy do you feel on a scale of 1 to 5, 5 being very happy and 1 being not happy.*1370

*Would have it mattered if they had set the scale from 0 to 4 instead?*1379

*1, 2, 3, 4, 5 versus 0, 1, 2, 3, 4.*1385

*You could see that if someone marks the 5 on the scale and some of them marks a 4 on the scale.*1393

*It is not that this person is less happy, there are the ones who are maximally happy, right?*1398

*It is just that they had a different scale that they were using.*1404

*These are examples of interval scales where the 0 actually does not mean 0 of happiness, *1409

*it is just whatever it is relative to the scale that you are using.*1416

*That is what we mean by no meaningful 0, you can often test for yourself whether something is a interval scale *1425

*by moving the scale a little bit and seeing if it is still okay.*1432

*If it is okay then you know you have an interval scale.*1439

*Let us say you get something like another survey question that says how satisfied are you with your job?*1440

*You will rate it on a scale of 0 to 100.*1447

*If it was on a scale of 100 to 200, would it make any big difference?*1452

*Not really.*1460

*That is how you know that it is an interval scale.*1461

*Finally we get to the crème de la crème, this is the highest level and if interval is missing a meaningful 0 I bet you can guess what ratio has.*1467

*Here we have order, we have even intervals, and we have a meaningful 0.*1478

*In case these are ratio scales are often things like height or weight where 0 means 0, none of something, none of some unit.*1491

*If you are 0 inches tall that means you are 0, that means 0.*1505

*That is the big difference between nominal, ordinal, interval, and ratio scales.*1515

*Let us look at some examples to exercise these concepts.*1523

*Here we have a preschool, elementary, junior high school, college and graduate school, form what kind of scale.*1529

*Let us see preschool, elementary school, junior high, senior high, college, graduate school, they have an order, check.*1538

*Is there even intervals? *1549

*The difference between preschool and elementary schools, preschool might take maybe 2 years and elementary school might take 6 years.*1556

*Even there along we could see they actually take different intervals.*1564

*Junior high might be 2 to 3 years, high school is 4 years, college 4 years, graduate school that could to be anywhere from 2 to 10 years right.*1568

*This definitely does not seem like they have even intervals.*1581

*And because of that even if we assign these things a number like 1, 2, 3, 4, 5, 6 it would not be that if we subtract that one it would be 0.*1588

*I would say there is no real 0 either .*1603

*Because it does have order, let us go with ordinal scale.*1607

*Example 2, in one state voters register as Republican, Democrat, or Independent, which scale of measurement is used?*1617

*Here is there an order to this like there was for the schooling?*1625

*Not really.*1630

*You may have a different opinion depending on your political leanings but these are just different categories of people.*1631

*I would say that this is a nominal scale.*1639

*Even if we assign numbers to it, they will be purely symbolic.*1641

*Example 3, a math professor gives students a 30 item test on the first day to ascertain his students basic math knowledge.*1649

*Bob got a 0, Joe got a 10, Carlos got 20 and Nate and Layla got a perfect score, what kind of a scale of measurement is this?*1657

*0 actually does sort of mean something if you think about it as how many items they got correctly.*1668

*And getting 1 item correct versus 2 item correct, this that ascertain their basic math knowledge?*1677

*Let us separate it out into first basic math knowledge.*1688

*Basic math knowledge is the actual variable that this professor is interested in.*1696

*Basic math knowledge is a continuous variable.*1703

*Somebody could have just a smidge more or just a smidge less than someone else so every value can be covered.*1707

*In order to get the values for this variable they used a certain kind of measurement.*1717

*He used a certain kind of measurement.*1725

*The measurement tool he used was this 30 item test.*1729

*The 30 item tests what kind of measurement scale is this on?*1735

*I would say it does have a true 0, 0 does mean something, you get 0 items correct.*1742

*It does have even intervals so when you are counting like how many questions correct and you know that 30 is better than 20 is better than 10.*1752

*It has order.*1767

*I would say that this is a ratio scale.*1770

*Just because it is a ratio scale does not mean that it actually measures basic math knowledge in a precise way.*1774

*After all someone who has a 0 on this test, it may not be that they do not know anything right so *1784

*how it actually matches that to the variable is still up for grabs as the question but in terms of the measurement scale it might be a ratio scale. *1792

*There is one way that it could not be a ratio scale and that is if the questions are differing levels of difficulty *1802

*so there are difficult questions and not difficult levels of questions, that could screw us up.*1814

*Let us just assume right now that all the items are sort of roughly similar levels of difficulty, if so then I would go with ratio scale.*1823

*Example 4, if the active measurement is disregarded which of the following variables are fundamentally discrete and which are continuous?*1834

*Temperature is probably continuous because you could be a little bit hotter, a little bit more hotter, a little bit hotter than that.*1845

*Every kind of value can we have on that scale, no gaps.*1855

*Time elapsed, this is also continuous because you could have every small increment of time accounted for.*1864

*In gender I would say this is discrete because there is not every single kind of variation in between.*1874

*Brands of orange juice, I would also say discrete this actually sounds nominal.*1886

*Size of family, this is also something that is discrete, again it is hard to have 2.75 people in the family.*1894

*Merit rating of employees so how much merit does an employee deserve?*1904

*Fundamentally that is continuous, one employee could be just a little bit better or worse than another employee.*1909

*They could be very close.*1916

*In the same way achievement score in mathematics that could also be continuous *1918

*because somebody might be able to achieve just a little bit more in math than someone else.*1923

*That is example 4, thanks for watching www.educator.com.*1931

*Hi and welcome to www.educator.com.*0000

*We are going to be doing a short lesson introducing you to Excel.*0002

*If you already worked with Excel before please feel free to move on.*0007

*Before we get to visualizing distributions in Excel, we just want to give you a little overview.*0013

*Excels are nice handy spreadsheet program.*0018

*It is pretty easy to use, most computers have it and it is useful because a lot of companies and laboratories use Excel.*0021

*It is a nice real life skill to have.*0029

*Another thing about Excel is that it is a good short intro to programming.*0032

*It can handle iterative computations, computations that you have to do over and over again and small calculations in bulk.*0036

*Here is how Excel is organized, it is based on workbooks.*0047

*Think of a file as a workbook, it is a series of what we call sheets.*0052

*Each file when you save an Excel file is a collection called the workbook.*0056

*Just to show you on a real Excel workbook, notice how it says workbook up there.*0065

*When you save this file and I hit save here, this whole file is going to save several sheets and the sheets are listed down here.*0070

*Now we only have one sheet but here I'm going to add on another sheet.*0079

*We have sheet 1 and 2.*0084

*You can have 4 or 5, all kinds of different sheet.*0086

*You can also rename these sheets to whatever you want.*0090

*We could call this one data.*0093

*And there you go, that is our sheets.*0097

*It is a little bit small here and let me try to, it still ends up being small but hopefully you could see that in the corner of your screen.*0099

*In each worksheet you are going to see columns and rows.*0112

*Columns are going to be shown to you and indexed by a letter.*0116

*Columns are always letters like ABCD.*0122

*The rows on the other hand are always going to be indexed by numbers like 12345.*0126

*Each cell or square has a name that you can index by saying the column name and the row name.*0132

*Something like A1, B5, these are all cell names.*0140

*Each cell can accept a number, text, or formula.*0146

*We will get into what those are.*0150

*Just to show you again in Excel here my columns indexed by letters like A, B, C, D.*0153

*Here my rows indexed by 1, 2, 3, 4, 5.*0163

*And each cell has A, B, C, D and 1, 2, 3, 4, 5.*0164

*If you click on this cell, this cell is B2.*0174

*Let us talk a little bit about the tools.*0183

*The toolbar in Excel usually have a menu bar which is sort of your standard Microsoft suite toolbar.*0186

*It also usually has a toolbar for things like formatting your words and letters, fonts, colors, whether you want things to be centered or not.*0195

*Those things are pretty basic.*0206

*It usually has a formula bar.*0208

*This is new to Excel and different from all the other Microsoft suite programs.*0210

*In order to let Excel know that you want to type in a formula, you start the formula with an equal sign (=).*0217

*Just to show you that on Excel, here I could write down A, B, C or I could write down a number.*0224

*Here we have your standard toolbar for things like hey I want to save it or I want to print it.*0237

*But then you would probably also have something like a formatting palette *0246

*to help you figure out what font you want to make it out, this 10 to be a red.*0250

*Do I want my 10 spaced in the middle or aligned to the left?*0259

*You can also make this 10 facing in different directions that we could turn it orthogonally.*0269

*Let me turn it back so that we will get to use it again.*0277

*If I want to write a formula I would just start by writing an equal sign (=).*0280

*A formula can take lots of things and we are going to into what some of those things mean.*0286

*One of the things that can do in a formula is I can reference another cell.*0290

*Let us say I want the cell to have whatever is in this cell B2.*0295

*If I click on B2 then this formula says this cell is going to be equal to whatever is in B2.*0300

*Click enter and it should have the same thing that was in B2.*0307

*I could change B2 like I could make that 100 and that is going to change this one immediately because it is just a formula.*0311

*It is just pointing to this cell and saying whatever it is in it, take that on as well.*0321

*Some of you may have a separate formula bar or you might by double-clicking in it be able to see what sort of written in here.*0328

*We will probably show it to you with the formula just type inside the cell but once again if you want to use the formula bar that is not a problem.*0337

*That is basically it for Excel organization, now we will go on how to reconcile Excel with the data organization *0356

*that we learned about in statistics so far.*0365

*Excel plus data, in Excel we know that a file is called a worksheet.*0370

*In statistics language that is where we are going to put our data.*0378

*Each row in Excel is referenced by numbers.*0381

*Each row in data is going to represent a case.*0385

*Whatever object we are interested in studying or analyzing.*0392

*In Excel the columns are going to be referenced by letters and these columns are going to represent variables in our data.*0398

*Each cell reference by a number and a letter step together like A1, that is going to take on a value.*0407

*One of our values goes into our variables.*0416

*That is how Excel and data come together hopefully you have learned a little something from the short intervention.*0423

*Do not worry if Excel is still a little bit new to you, you will get used to it at the end of this lesson.*0430

*Thanks for using www.educator.com.*0436

*Hi welcome to www.educator.com.*0000

*We are going to be talking about how to create frequency distributions in Excel from raw data.*0003

*We are just going to overview when sample data set in Excel already, you can download it from one of the links below.*0012

*When we are going to talk about how to create frequency distributions from that data *0022

*but in order to create these distributions visualize a bowl of seeable distributions.*0027

*We need to go first from the data to frequency tables, then from the tables we will go to the visualizations.*0034

*First, going from raw data to frequency tables.*0046

*The reason we want to do this is oftentimes when we look at raw data it is really hard to make sense of.*0050

*It is just rows and rows and rows of data.*0055

*It would be nice if somebody could summarize that data for us so that we can visualize it.*0059

*When we summarize and visualize that data we get a sense of what the data looks like.*0066

*We are going to be talking later about actual shapes of distributions.*0071

*There are two ways to go and do frequency tables in Excel.*0076

*One is by using formulas.*0080

*Here we are going to be using the formula count F and the other way is to use pivot tables.*0083

*I’m going to show you one example of using pivot tables but we are going to be using mostly the formulas.*0090

*If you want to open up your Excel file that has all of our data in it, this is a sample data set of 100 friends from www.facebook.com.*0100

*Notice that they all have this CID which is their case ID and each column shows some sort of characteristic or variable.*0110

*Each cell for each person has a value for that variable.*0124

*Let us look at example 1, CID 1, case number1.*0131

*For this person they have 4 tagged photos, not a lot of tagged photos.*0137

*They have to seem 0 mobile uploads, again not a lot of mobile uploads, maybe they do Not have a smart phone right?*0143

*If we go down the line we could see that there are lots and lots and lots of variables here.*0150

*There are tagged photos, mobile photos, uploaded photos, profile pictures, then number of friends, number of siblings, relationship status right?*0154

*There is a whole bunch of these.*0167

*Here is one that we are going to be focusing on today, birth month.*0169

*Birth month is going to be important for us today.*0172

*We are going to be looking at age and height.*0176

*If I asked you if you see these 100 people and I will show them to you all at once so you could see them.*0184

*Here is this 100 people what can you tell me about their age.*0193

*What can you tell me about their height?*0197

*It will be hard to do because it is just lines and lines and lines of data.*0199

*It will be nice if there was one way where we could just easily see all the data at once in a way where it was a little more tangible to us.*0204

*That is where we are going to be talking about how to visualize these and how to create frequency tables.*0216

*In the files that I provided for you, I put in little tabs already.*0219

*One of the sheets has all of our data in it and one of the sheets talks about the variables.*0225

*Here we have a whole bunch of different variable names like the case ID number, the tagged photos, *0235

*how many photos they are tagged in, mobile uploads, how many mobile photos uploaded, relationship status, birth month, birth year, gender.*0239

*These are a whole bunch of different variables that are already in this data set.*0249

*I also have a column that tells you what kind of measure it is.*0254

*Is it a nominal measure where it is just a number but it really stands for a name?*0259

*Relationship status is one of those where there is a number there like 1, 2, 3 or 4 but it does not mean *0264

*that the relationship status is literally like the number 1, it actually means if you scroll over, if they have a zero it means that their 0440.9 relationship status is blank.*0271

*If they have a 1 it means that their single.*0283

*If they have 2, that means there in a relationship.*0286

*If it is 3, they are engaged.*0289

*If it is a 4, they are married and if it is a 5, it is complicated.*0291

*And 6 if it is other right?*0295

*That is an example of what we call a nominal type of measure.*0297

*Just so you can see all of these things at the same time, if you look down here there is this two little blue rectangles.*0302

*If you drag that over then you could sort of keep this column just static and locked while you move these columns.*0311

*We can also see that birth month is what we call an interval, it can also be seen as ordinal.*0324

*It is not quite interval because it is technically like 30 or 31 days, it is not exactly the same interval but you could sometimes call it interval.*0332

*Each of the numbers represent one of the months.*0344

*Birth year is also interval, there is an interval of exactly one year.*0350

*Gender is obviously nominal because even though there is a 1 or 2 it does not mean that their gender is 1 or 2.*0354

*It means that if they have a 1 they are male.*0362

*If they have a 2, they are female.*0364

*Some things like friends is really to understand though because friends is a ratio measure.*0366

*It is the count of how many friends they have so that is continuous type of variable and if they have a 0 means they have no friends.*0372

*That is very rare on www.facebook.com but it could happen.*0382

*I’m going to move this locked piece over.*0387

*The next tab you could see there it says birth month on it.*0391

*So far I have created a little set up so that we could begin our frequency table.*0397

*A frequency table is just a count of how many people are born in January.*0402

*How many people are born in February and so on and so forth. *0408

*Now if we have to do that by hand it would be hard.*0412

*We have to go to our data, click on data.*0414

*Go to birth month and we have to count up how many people have one, 1, 2.*0418

*But this is a very error prone process so we are going to use Excel to help us do that really efficiently.*0426

*First, let us go to our first example.*0436

*We have here a data set with data from 100 www.facebook.com friends.*0440

*More of these friends born in a particular month or is the number of births fairly uniform across the year.*0444

*Well is there reason to believe that one month is more popular for having babies than another month?*0452

*We are not sure but it is hard to see the answer to this question literally like see the answer to this question *0458

*by looking at the data because the data just look like this giant list.*0464

*That is why we are going to create frequency tables.*0470

*In order to create frequency tables we can start off with the formula.*0474

*In order to do a formula remember we always start off with the equal sign (=) to tell Excel “hey I’m doing a formula here”.*0479

*In order to count how many ones we have we could use the count formula.*0486

*It is a formula that is already prewritten in Excel.*0493

*Excel will just do it for us.*0496

*If we just stopped at the word count, it would just count how many things you have.*0498

*It would not count how many ones you have, right.*0504

*We want to use the formula count if, that is the function that we want to use*0508

*What is handy about Excel is that once you type in something then it will tell you what inputs you need.*0514

*Here it says you need the range.*0521

*The range of cells that you want Excel to look at as well as the criteria.*0523

*Here I’m going to tell Excel we will look over at my data.*0529

*I’m going to click on data and click from this one all the way down to the very very last row. *0535

*And if I go back to birth month then it should say date from row I2 all the way to I101 but it has it twice, I’m going to delete this part.*0547

*That is the data that I wanted to look at.*0567

*This little column right here is telling you the range.*0570

*It says go from I2 all the way to I101.*0574

*That is the criteria I want and before I put my criteria Excel tells me, it reminds me I need a little comma in between.*0579

*I’m going to put a little comma.*0589

*What is my criteria? I wanted to count it if it is a 1.*0591

*I’m going to say if is equal to whatever is in this cell.*0598

*Excel will automatically put in that this is part of the birth month sheet.*0602

*It actually does not need this one either but it will put it in automatically for you.*0611

*I’m going to delete that one just so you could see but you could have it there as well.*0616

*It does not matter.*0620

*Let me finish my little function and let us look at what it says.*0623

*It says count if the data in this range is basically equal to whatever is in a2, this one.*0626

*Let me hit enter and it should say 7.*0636

*7 people out of my 100 www.facebook.com friends are born in the month of January.*0639

*The great thing about Excel is that it is a relative program.*0645

*If I copy and paste this cell, one cell down it will take everything in my formula and sort of calibrate it one cell down, right.*0649

*Let me look at this, do I wanted to bring everything one row down?*0665

*That means my data would go from I3 to I102.*0671

*That is not what I want.*0677

*I want the data part to stay the same but I want this part to move and moved down.*0678

*So that then it will say count if this data is equal to 2.*0684

*Here is what I’m going to do, to tell Excel keep this part the same.*0691

*I’m going to tell I’m going to put in a dollar sign ($) right in front of the I and right in front of 2.*0695

*This says freeze the row and freeze the column.*0702

*I’m going to put that also in front of this one, as well as that one.*0706

*That means this data set will never move but this A2 will move.*0712

*Notice that doing that does not change anything from my first row but I’m going to take this and copy it.*0718

*I’m just hitting either command c if you are on a mac and control c if you are on a pc and then pasting it one cell underneath.*0724

*Let us double click on this to see what it says.*0736

*It says count if data and my data states exactly the same from I2 to I101.*0739

*That is exactly what I wanted to do.*0745

*Notice that now my criteria has changed.*0747

*My criteria has moved one row down because I have copied and paste in my formula, one row down.*0750

*Excel it is relative.*0757

*It will move everything one row down.*0759

*Let us try it with the next one.*0762

*I’m just copying and pasting this one, one row down.*0764

*Let us double click on it to see what it says.*0769

*It says count if.*0771

*Data stays exactly the same from row 2 to 101 but now it is comparing it to whatever is in A4 which is March.*0774

*The nice thing about Excel is that if you look right at the corner here, there is this little box in the lower right hand corner.*0785

*If you put your mouse over that it will turn into a little cross.*0794

*If I drag that all the way down, it will copy and paste my formula again and again all the way down.*0800

*We could just check one of these down here once again my data set has stayed the same because I put those dollar signs ($) in there.*0807

*My criteria has moved down to A10 now.*0816

*I have my frequency table now.*0820

*Frequency tables are nice because they just give you the raw numbers in the month of January there are 7 people who have birthdays then.*0824

*In the month of July there are 10 people who have birthdays then.*0833

*We could look at our data.*0837

*We could stop here but I want to show you another way that we could create frequency tables.*0839

*I’m going to go back on my data and show you a second way.*0848

*The second way is less common but I still want to show it to you because we may use it once in a while.*0853

*We are going to use what is called pivot tables.*0857

*What I’m going to do is just put my cursor anywhere and open my Excel toolbar.*0862

*Unfortunately, you cannot see it on this screen.*0870

*Open my Excel toolbar.*0872

*There is a little tab called data.*0873

*Seldom used.*0877

*If you scroll down there should be something that says pivot table or pivot table report.*0880

*I’m going to click on that.*0888

*Once that comes up, you should have a little pivot table wizard that pops up and you will say “where is this data you want to analyze?”*0893

*It is on my Microsoft Excel data base.*0903

*Is this the data you want to use?*0907

*Yes, I want from A all the way to N and from A1 all the way to 101.*0908

*That is next.*0924

*I want to put my pivot table on a new sheet, just so I can show you.*0925

*I’m just going to hit finish.*0930

*A new sheet should pop up, it is probably be called sheet 1.*0934

*I’m just going to make this a little bigger for you.*0939

*A little pivot table should pop up.*0945

*You should also have a little pivot table tool bar that also pops up.*0949

*Let me drag it in for you.*0955

*Here we go.*0966

*This is the little pivot table tool bar that comes up.*0967

*This pivot table tool bar has all of my variables in it.*0970

*I could drag these variables into this pivot table down here.*0975

*It actually shows why it is called a pivot table.*0979

*I assume it is because you could move these variables from one corner to another and that is where we get the pivot.*0982

*What we want is a bunch of months on this side and then I want it to tell me how many people are born in that month.*0990

*I’m going to look for birth month and put it in my row fields because each row is going to be a birth month.*1000

*I’m going to take that birth month and drag it into my data as well.*1006

*What is does is it sums up how many of those birth months there are.*1012

*For January it sums up 1, seven times but for 2 I do not want it to sum up.*1018

*Instead I’m going to tell my pivot table count how many they are, do not sum them up.*1026

*Go to pivot table and go to field settings and I will hit count instead of sum.*1032

*Then hit Ok.*1039

*When that happens you can see we basically get the same numbers that we have when we use the formula.*1040

*In the month of January we have 7.*1045

*In the month of July we have 10.*1048

*This is another way that you could look for frequency tables.*1050

*Notice that this one is pretty fast.*1056

*Pivot tables do require a little bit of work but on the front in there is a little bit of learning curve.*1059

*Once you do understand that, they are really handy.*1066

*We maybe using them again in the future.*1068

*If you do not feel comfortable with them, feel free to also use the formulas.*1072

*I will be using the formulas for the rest of this lesson.*1076

*Let us go back to my birth month.*1079

*My birth month pivot table created just through Excel formulas by themselves.*1083

*I have this nice frequency table but it will be nice if I could visualize it.*1089

*Here I have to read each row and although for 12 months it is not so bad, they might be times when this is less helpful to us.*1097

*What I’m going to do is highlight the data that I want to visualize and then hit chart.*1105

*It should be one of the tabs up here or you could go get it through one of your Excel tabs.*1115

*I’m going to say give it to me in columns or you could use borrow as well.*1122

*In Excel it just means it is on this side.*1133

*I’m going to use columns for now.*1135

*I will just pick the first one.*1138

*It seems the simplest.*1140

*I’m just going to delete that legend, it is redundant.*1144

*Here is my frequency table and we could literally see our data.*1149

*It is also tells me what each of these bars stand for.*1157

*It stands for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.*1162

*What Excel will do is it will automatically seed your X axis with just those numbers starting from 1 and it will go up.*1166

*With months, handle it but is the same thing that Excel is doing.*1174

*In another example we will need to put in our own X axis.*1179

*Notice that here these are not means, they are not averages, these are frequencies.*1185

*This means that 7 people were born in January, 10 people are born in July and 7 people are born in December.*1191

*And so that is what our birth month frequency visualization looks like.*1205

*This is our frequency distribution for birth month.*1210

*Let me minimize this.*1215

*If the number of friends born in a particular, is one month particularly popular for one of our friends are born.*1217

*It does not seem to be the case, the months all tend to be from something like 7 – 10 people per month.*1225

*It seems that the numbers are fairly uniform.*1233

*Let us go into our second example.*1241

*Here is another example and now let us take our same data, the data from 100 www.facebook.com friends *1243

*and we are going to look at what is the age distribution in this sample.*1249

*Here is my Excel data I'm just going to click on the data sheet and here when we go up and look for age we could see here is a whole bunch of ages.*1255

*It seems like there is a lot of people in their 20’s.*1268

*A few people in their late teens but here we see some people who are 0 years old.*1274

*In this data set, if they have 0 it means that they do not list their year of birth or do they do not list their age.*1281

*Maybe they are embarrassed, maybe they are too young.*1289

*I do not know.*1291

*We do not learn a lot by just scrolling up and down on this data.*1296

*That is why it will be nice if we could look at a frequency table or look at a distribution visually.*1300

*I'm going to click on my age sheets and here I have already made set it up so that we could just *1306

*do our frequency table really easily from the lowest age in our sample which is 17 I have ignored the 0 obviously *1314

*to the oldest age in our sample which is 38 and there is all the ages in between.*1323

*Let us go ahead and put in our formula to find out how many people in our sample are 17 years old.*1329

*To start a formula we start with the equal sign (=).*1335

*We use count if because we do not want to count everybody, we just want to count the people who are 17.*1338

*Let us tell Excel where it should find our data, what is the range of data.*1346

*I’m going to click on data and click from this cell all the way down to row 101.*1352

*I know I need a comma after that.*1364

*I’m going to delete that part.*1371

*Here is our data range and I wanted to count it if this person is 17.*1373

*My inputs are there. Remember we want this data to stay the same all the time.*1385

*We do not want it to move because Excel will move it if it has the chance to.*1389

*I’m going to put dollar sign ($) in front of the L and the 2 and in front of the column indicator *1394

*and the row indicator to tell it to lock this data in place.*1400

*Always use this data, do not change.*1404

*Once I have that formula, I'm just going to drag it all the way down so that it counts at the frequencies for 18, 19, 20 year olds.*1409

*Let us back check. Let us look at 21 year olds.*1424

*It says count if our data set has stayed the same because we have locked it in with our dollar sign ($).*1426

*Now it is saying I will count these people if they are 21 years old, that is our criteria.*1433

*It looks like our formula has copied and pasted quite well.*1438

*Notice that for some of these some of ages, the frequency is 0.*1443

*There are 0 people who are 26 years old in our sample.*1449

*Now why do I want to keep that 26 in there?*1454

*If we skipped down on 26 and 28, 29, 30, 31, 32 and we looked and there is 127 year old in 133 year old, *1457

*we might mistakenly assume that from 27 to 33 there is equal chance of having *1471

*at least one person from our sample being sort of in that range.*1478

*You could see that is actually not true.*1483

*In between there, there is like a big desert of nobody and we want our distribution to reflect that.*1486

*Age is a continuous variable and so we do not want to skip any ages.*1494

*We want to show how the distribution looks as we look at age continuously.*1500

*This is nice because we can already see that the ages are clamped or clustered around age maybe 20 – 22, early 20’s.*1507

*It will be nice if we could really look at this.*1518

*One thing you might want to do is click on select both age and frequency.*1520

*Go to charts and we are going to do an X, Y scatter.*1530

*For those of you who have Microsoft Excel later than 2008, like 2009 and later you can go directly to column *1538

*but here we are going to start with 2008 Excel.*1549

*We are going to need to do a little fix.*1555

*First I’m going to click on a scatter.*1557

*A scatter is nice because it shows you both the age.*1560

*This is age 17 and the frequency.*1566

*Once we have that then I'm going to go to column and then it will show me 17 through 38.*1572

*If I had gone directly to column, here is what will happen.*1584

*If I did not go through scatter first, here is what will happen.*1589

*Let us say I just wanted the frequency, they will go directly to column it will not give me the proper ages on my X axis, *1595

*it will only give me Excel’s default setting for the X axis which is just labeling it from 1 all the way to 22.*1604

*However many there are that is not what we want.*1614

*Instead we would rather have Excel label the correct ages for us.*1618

*Just so that we will know that this is a frequency distribution of ages later.*1625

*We should go and label are horizontal X axis, we can label that age.*1632

*In that way we will know it is a frequency table but it is a frequency table of ages that is what the 17 stands for.*1641

*What is the age distribution in the sample is largely young.*1654

*They are mostly on the young side with a few people sort of in their 30’s.*1658

*Example 3, again from our same www.facebook.com data, what is the height distribution in this data?*1666

*What did their heights looked like.*1673

*Let us see.*1675

*If we click on data and we look at their heights, their heights are listed in inches.*1679

*Remember that 5 × 12 is 60, 60 inches is about 5 feet tall and then 68 is 5’8.*1685

*It is a quick way to think of it.*1696

*72 is 6 feet tall, that person is pretty taller.*1698

*Once again if we just look at these row by row, it is a just bunch of numbers.*1703

*We do not need that, we would rather have a nice frequency table.*1709

*Let us go to height.*1713

*I have already seated it for you with the height that is the minimum height in our data set as well as the maximum height in our data set.*1716

*The minimum height happens to be a little bit just shy of 5 feet, 4’10.*1724

*This one is a little bit more than 6 feet tall, 6’3.*1734

*Let us put in our frequency function.*1740

*Count if and let us go ahead and select the data that we want to use.*1746

*Now that we know we basically need to lock it in place, let us do that right here.*1759

*Let us lock it in place.*1766

*We already locked our data in and what is our criteria?*1774

*I want you to count it if they are 58 inches tall.*1779

*It seems that there is only one person in our data set of 100 that has that height.*1787

*I’m just going to copy and paste that all the way down.*1792

*Once again I'm just going to spot check, 69 inches tall count if this is the correct data.*1796

*It is locked in and this is the correct criteria that I wanted to use for that row.*1806

*Good.*1811

*When we look at this, it seems that it is not that there is one cluster.*1814

*It seems like there is this sort of giant spread out cluster.*1818

*It will be nice if we can look at this visually.*1825

*Let us go ahead and select both columns.*1829

*Go to chart and go ahead and select XY scatter.*1833

*This is going to give us both, it is going to use the height as the x coordinate and the frequency as the y coordinate.*1839

*Here we see that all our frequencies are up here because all of our heights are from 58 to 75 inches.*1850

*Let us change that into a column chart.*1862

*Here is how our distribution looks like.*1870

*Just in case we come back to this later it will be nice to know what these numbers down here represent.*1874

*I'm going to go to my formatting palette, I’m going to close that.*1879

*I’m going to go to my formatting palette and tell my horizontal axis that it should be labeled height in inches.*1884

*That is what our distribution of heights looks like.*1902

*It looks like these over here, this one seems pretty popular and these seem sort of popular.*1907

*These are less likely and this one a little bit less likely.*1915

*This is a sort of what our shape looks like and it is nice and it is really easy to see when we see it in a visualization.*1921

*It is harder to see when we just look at the list of numbers.*1928

*Let us move on to our next example.*1934

*Example 4, now that is the height distribution of everybody in our 100 person www.facebook.com example.*1940

*But it is a mix of males and females.*1948

*What if we just wanted the height distribution of males?*1951

*After all males tend to be taller than females.*1954

*Their distributions might look different.*1956

*Let us look at the height distribution only of males.*1958

*We could also look at only the height distribution of females.*1962

*Feel free to do that if you want.*1965

*Here I'm going to use my height by gender and there is a male frequency column and a female frequency column.*1970

*Once again here are my heights but we will have to figure out in our data set which rows belong to males and which rows belongs to females.*1982

*Let us go back to our data set.*1993

*Here is my column for gender, my variable of gender.*1998

*Some people are gender number 2 and some people are gender number 1.*2003

*If we look at our variables we could see that gender has been dummy coded because it is a nominal measure.*2008

*We will get 0 if gender is blank or unavailable.*2019

*They got 1 if their gender was male and 2 if their gender was female.*2024

*Here is what we will do, we will take all of our data and sort it by gender *2030

*so that all the 1 are clumped together and all the 2 are clumped together.*2035

*I'm going to use sort.*2041

*Sorry about that.*2054

*I think I did it and ended it, alright.*2059

*I’m going to use gender and I’m just going to sort it by clicking in this column.*2060

*I just want to make sure that these guys all moved with each other.*2070

*Now it is sorted so that all of my data for males is up on top and then all of my data for females is at the bottom.*2077

*Just to keep it straight for myself, I’m going to just color all the heights of males, all the values for height of males,*2088

*I'm going to color that with the blue font color.*2098

*Just to help myself keep it straight I’m going to color all the females height values with the sort of pinkish font color.*2106

*What does my distribution of only males look like?*2119

*We need to start off with the frequency table again.*2123

*Let us go to height by gender and here I will put in count if.*2126

*And let us put in my range.*2136

*Now my range is only going to be those that I have already colored blue *2138

*because they only want my range to be those that are already identified as males.*2143

*Here I’m going to select all these blue guys and put a comma.*2150

*And then tell if a male is 58 inches tall then I definitely want you to count him.*2164

*It turns out there are 0 males that are that tall or that short for that matter.*2178

*We want to lock that data set in place because we know that this is not going to need to move for this column at least.*2184

*I’m going to go ahead and copy and paste that all the way down and we see that *2195

*from the males the heights are sort of clustered up here rather than down here.*2200

* I wonder if that is the same for females.*2208

*Even though our question was really about males why do not we females too just to see.*2210

*I’m going to start with my count if.*2217

*The range for females needs to be all the data that has been already identified as females.*2221

* Here are these pink women and I’m going to go ahead and put in a comma because I know I will need one.*2227

*Go back here and I will say check if the female is that height.*2235

*Once again I want to lock in my data.*2243

*I do not want that to move when I copy and paste.*2248

*And then it turns out that our one person who is 58 inches tall before happened to be female and I’m going to drag that all the way down.*2253

*We see something different in females than we saw in males.*2263

*Females tend to be clustered around here and the most frequent height being about 64 inches.*2267

*For males, the heights are sort of clustered up here with the most frequent height being 69 inches.*2275

*Let us look at this now and visualization.*2283

*I’m just going to look at the heights of males for now.*2288

*Hit chart and go to XY scatter because I want to know both the height and the frequency of that height.*2293

*We see that males are clustered up here.*2305

*Let me change that into a column and what do we see?*2309

*We see that it is like a pile.*2318

*The males are sort of piled up around 68 – 70 and it falls off closer to 5 feet tall.*2321

*There is not as many people who are way taller than 6 feet.*2330

*That is the chart for males.*2337

*Feel free to go ahead and do the chart for females.*2340

*That is the end of our examples today.*2346

*Thanks for using www.educator.com.*2348

*Hi and welcome back to www.educator.com.*0000

*We are going to be talking about frequency distributions again but now we are going to be going a little more into detail about their features.*0003

*In the last lesson we covered how to look at the data in Excel.*0013

*There is a checkmark on top of that one and we talked about how to go from data to frequency tables using our count if function. *0017

*From frequency table to visualization.*0026

*We are going to take another look at those same examples that we looked *0029

*at before except now we are going to be talking about the features of these distributions.*0033

*In particular we are going to be looking at their shape.*0037

*There are couple of shapes you should know after this.*0040

*One is uniform distributions.*0043

*Another one is going to be called unimodal.*0047

*Yet another is called bimodal.*0054

*Especially we are going to be looking at called normal.*0059

*We are also going to be talking about center.*0065

*We are not going to be talking about how to calculate the center of the distribution.*0068

*We are going to be talking about how to think about the center conceptually in three different ways, mean, median and mode.*0072

*We are not going to talk about how to calculate it yet.*0080

*We are also going to be talking a little bit about spread.*0084

*How spread out is this distribution.*0086

*Finally we will also mention outliers, gaps and clusters whenever they are relevant.*0090

*Recall example 1, here we looked at a data set of a 100 www.facebook.com friends *0098

*and we looked at whether more of these friends are born in a particular month or another.*0103

*Note here that it really seems to be that no particular month is super popular.*0108

*This is what we call the uniform distribution.*0113

*If you sort of squint and blur your vision a little bit, it is almost like there is a flat line here.*0116

*Everybody is hovering close to that line.*0125

*No one month is more frequent in births than any of the other months by a lot.*0130

*Some of these months are a little more frequent but only by a little bit.*0136

*You could see there is relatively little change from month to month here.*0143

*Other uniform distributions also look like this sort of rectangle or flat shape *0147

*and these distributions might be anything from deaths occurring on days of the week.*0155

*Is there any reason to believe that one particular day is more favorable to die on than the other?*0160

*Or in rolls of a six sided dice, is there a particular reason to believe that one side might come up more frequently than another?*0165

*Not if it is a fair sided dice.*0174

*Remember this is now example 2, in example 2 we look at the same data set again and we looked at the age distribution in the sample.*0181

*Here we do not have a uniform distribution.*0191

*No matter how much you squint your eyes you are not going to see sort of a flat shape.*0193

*You will see a peak right here and because of that, this peak often called a mode the most frequent value.*0197

*This peak makes this a unimodal distribution.*0208

*I’m not going to call it example 2 anymore, I’m going to call it a unimodal distribution.*0213

*We will add on to that.*0220

*Not only that but this shape is what we call skewed.*0222

*If I decide to just draw a light little sketch over this guy, we see that it has this long what we call tail.*0226

*This tail goes out towards the right side, the larger values because it is skewed and the tail is to the right we call is skewed right.*0242

*It is not only unimodal but it is also skewed right.*0256

*You often have a skewed distribution when you have some sort of minimum or maximum value that these values are all bumping up against.*0263

*In www.facebook.com I think you have to be 13 years old to sign up and maybe a lot of 14 and 15 year olds.*0273

*Their parents are not letting them sign up.*0279

*The bottom end of it there is sort of a walls and there is like an imaginary wall there.*0282

*The most popular at least in our sample seems to be in the 20’s and some of the older people use it.*0290

*There is no limit on that.*0296

*You could be 100 years old and still use www.facebook.com.*0297

*Since there is no limit on that, that tail can go on for a really long time.*0300

*These outliers out here, you could think of them as oddballs but we call them outliers.*0306

*Tails are often made up of outliers.*0320

*Note also that because this is skewed right, if we drew a line of symmetry from the mode and *0324

*we imagine folding this distribution on itself, we would not have two sides that match up.*0333

*We call this asymmetric as well.*0342

*We learned a lot here, it is unimodal, it is skewed and it is asymmetric.*0346

*Here we will learn yet another term, we see there are these gaps.*0356

*These are called gaps, nice and easy.*0365

*If we had a couple of people clustered in a group we call that a cluster.*0369

*A lot of these terms are pretty normal words that you use in everyday life.*0377

*Let us move on to example 3.*0391

*In example 3, we are interested in what the height distribution was in this sample.*0393

*Compare this distribution to our previous skewed distribution.*0401

*Is it skewed to the right or skewed to the left?*0408

*Is there some sort of tail here?*0411

*Not really.*0414

*There is no real tail that I can see but we do see that there are a couple of places that are popular modes, most frequent values.*0416

*These are 64 and 69, these seem to be the popular peaks.*0428

*Because we have one mode here and another mode here this is no longer a unimodal distribution.*0436

*This is what we call a bimodal distribution.*0443

*Instead of calling it example 3 I’m going to call it bimodal.*0447

*Is it symmetric? We could see it as almost having 2 bumps like that.*0453

*It is sort of symmetric but not perfectly symmetric.*0464

*There is no tail, there is not very many gaps.*0469

*There is maybe a little bit of a gap here but not very much.*0476

*This is what we call a bimodal distribution.*0480

*Let us think about this, height distributions.*0483

*Well our www.facebook.com friends are both males and females.*0488

*and since males tend to be taller than females on average it might be that there is a cluster of males up here *0492

*and a cluster of females down here that we cannot see right now.*0499

*Let us look at these two distributions, males and females separately.*0504

*Here is just the distribution of male heights from our sample.*0511

*Notice that here it is not really a symmetric because when you look at this mode, there is our mode right here and you draw a line of symmetry.*0517

*You imagine folding it on itself then you will get a pretty even looking hill right there.*0531

*You will get a pretty even looking hill with roughly similar numbers of people on this side as on this side.*0547

*This is what we would call a roughly symmetric distribution instead of example 4a.*0558

*It is also what we call unimodal because we only have one mode right here.*0568

*What else do we notice about this?*0582

*We do not really see a tail and further more it seems that this distribution seems to have a lot of people piled up around 69 inches.*0584

*With a lot more people close to 69 and fewer people farther away from 69 like at 75 or around 64.*0596

*This is what we call a normal distribution.*0610

*You could think of a normal distribution like a pile.*0617

*A normal distribution will not usually, by definition a normal distribution is both unimodal and symmetrical.*0620

*In a normal distribution typically the mode, as well as the mean or the average is going to be the same.*0632

*To think about the word average you might want to think of it like this in terms of distributions.*0648

*Imagine cutting out this distributions, like out of cardboard and then trying to balance it on your finger.*0653

*Where the distribution would balance, that point, that is the mean.*0662

*Although we will learn to calculate this later, that is the image I want you to think of when you think of the mean.*0668

*If we draw a smooth line around this distribution on either side of the mode, at about 60% of the height of this peak that is about 50%.*0675

*Around here at about 60% of the height of this peak you will have what is called the point of inflection.*0692

*Here is what so important about the point of inflection.*0709

*Although you cannot see it very well from my picture, I will exaggerate it.*0713

*The point of inflection is where the distribution goes from being concave to being convex.*0716

*That is about right and this point of inflection is going to be important later because this distance right here, *0726

*this distance is going to be called the standard deviation.*0736

*Later we will learn exactly how to calculate that but that point of inflection and the standard deviation *0752

*is going to be really critical to our understanding of other distributions as well.*0758

*Here we see both males and females heights plotted on this frequency distribution.*0765

*Just you could see here is a sort of our female distribution and here is our male distribution.*0777

*Here you could see there is roughly a normal distribution for the females as well as the males.*0792

*We are going to say two normal distributions.*0799

*They are both unimodal and they both are roughly symmetric on both sides.*0806

*There is no tail.*0814

*No big gaps.*0816

*There is a big cluster in the middle and that is about it.*0817

*Here what we thought before was a bimodal distribution we actually see there is actually two normal unimodal distributions instead.*0823

*Let us summarize what we have learned so far. *0839

*We have learned four different shapes, uniform, skewed to the right or to the left, bimodal and normal.*0843

*We have also learn asymmetric and symmetric.*0850

*Here I’m asking is this one symmetric or this one asymmetric?*0853

*The uniform one, yes it is largely symmetric because rectangles are symmetric.*0858

*Skewed, are they symmetric?*0863

*No, because either the right tail was long or the left tail is long.*0866

*Bimodal, are they symmetric?*0878

*This one is sort of a sometimes.*0881

*There can be times when the these are symmetric.*0883

*For instance if you have two that look like that, it is roughly symmetric but you may also have bimodal distributions that look like this.*0886

*Then that one does not look as symmetric.*0897

*Normal distributions, yes they are symmetric, always.*0901

*I will just draw this here just so that you know.*0907

*Let us talk about the centers.*0910

*Does it have a clear mode?*0913

*Here it does not have a mode, there is not one most frequent value.*0917

*In fact all the values are roughly similarly frequent.*0923

*We will say no, it does not have a clear mode.*0926

*Typically the skewed distributions are unimodal.*0929

*Yes, unimodal.*0933

*What about bimodal distributions?*0940

*Do they have a mode?*0942

*Yes, they are overflowing with modes.*0944

*They have two modes in fact sometimes more.*0944

*You could have trimodal right? Yes.*0949

*What about normal distributions.*0954

*Well of course it has a mode because it is also unimodal.*0956

*Let us talk about spread.*0966

*What a spread look like here.*0968

*The spread is roughly even but as it goes as far as the values go.*0970

*Does it use the point of inflection? No.*0976

*What about in a skewed distribution?*0980

*Do we use point of inflection there?*0982

*In a skewed distribution the point of inflection is weird because the point of inflection is going to cut it up *0985

*at different places depending on whether you look at the right side of the mode or the left side of the mode.*0990

*Point of inflection is not quite as useful here.*0995

*In a bimodal distribution sometimes you can use the point of inflection but it gets complicated.*1000

*We will write in it is complicated.*1005

*It is only for the normal distribution that the point of inflection comes in really handy.*1014

*Resulting yes.*1019

*At the distance from the mode or the center to that point of inflection, distance is called the standard deviation.*1021

*Let us go on to some examples that you might frequently see in text books, AP statistics, as well as a lot of general reasoning questions.*1042

*These are what I like to call sketch problems.*1056

*They will give you some sort of data set that you only know a little bit about and they ask you what kind of distribution do you think it might have.*1059

*We can answer these questions now.*1069

*Here is sketch problem number 1.*1071

*What if you are asked to imagine the age of each person who got his or her first drivers license in your state last year?*1074

*That is going to be a distribution.*1081

*It is a whole bunch of numbers, whole bunch of different ages.*1083

*Let us think about this.*1088

*On the X axis we will probably put age.*1091

*Here we are going to put frequency but I'm not going to try that in.*1094

*Actually I will try that in.*1106

*Here is the Y axis.*1108

*Let us think about this.*1110

*Is there some sort of minimum or maximum age at which you can get your drivers license?*1112

*Yes, probably 16 in most states.*1116

*We will put 16 as the minimum age and probably a lot of people get their drivers license sort of early on, from 16 to 20.*1122

*They are probably very few people getting their first drivers license ever by the time they are 25, 30, or 40, even fewer people.*1131

*That is already starting to sound like maybe somewhat of a skewed distribution.*1143

*Probably lots of people in their early 20’s, maybe late teens, getting their drivers license *1149

*and very few outliers were getting their first drivers license when they are 40 or 50.*1158

*Even though you might not know very much about people getting their first drivers license you can already tell the shape of this distribution.*1168

*It is skewed but not only skewed but the tail is to the right.*1178

*We call that skewed right, it is probably unimodal or there is probably some cluster up here.*1182

*It is probably asymmetric because it is skewed.*1191

*Next example, here is sketch problem number 2.*1200

*Let us think about the life expectancy of females in Africa and Europe.*1205

*When we think about life expectancy that is considering how long are females in Africa and Europe going to live.*1211

*Age or years should probably be on the X axis.*1217

*On the Y axis once again we are going to be looking at frequency.*1224

*I will just say freq.*1228

*Let us consider the life expectancy in Africa and Europe.*1232

*Africa has a lot of diseases and malnutrition and other factors that are going to affect life expectancy of females.*1236

*Also Europe on the other side of the spectrum is going to have a lot fewer of those same issues.*1245

*The life expectancy of males in Africa might be shorter than life expectancy of those in Europe.*1255

*We might see something like a bimodal distribution that is actually caused by two unimodal distributions.*1261

*Let us put Africa in red and European females maybe in blue.*1269

*Maybe most European females die when they are older, like 70.*1281

*Maybe in Africa the life expectancy is less, maybe 50.*1288

*Here we see two unimodal distributions but it did not ask us to plot this separately.*1298

*When we combine these, we see a bimodal distribution.*1305

*Let us go into the next problem.*1319

*Sketch problem number 3 says well what about the distribution of the last two digits of the telephone numbers in the town or city where you live?*1323

*Do we have any reason to believe that those two digits are going to be more favorite than the others?*1332

*Let us think the last two digits of the telephone numbers.*1341

*If we put that on the X axis basically we can go from 00.*1345

*We can go from 00 all the way up to 99.*1362

*That is our range of possibility.*1366

*Let us see what might the frequency be.*1371

*The only we have a reason to believe that 00 is more or less popular than 99.*1375

*We do not really have a reason to think 99 is more or less popular than 62 or 47 or 35.*1381

*We might be thinking about a roughly unimodal distribution where each of these are roughly equivalently popular.*1390

*You can continue that on, so this is probably one of those unimodal distributions where one of the numbers is way more frequent than another one.*1405

*Let us move on to sketch problem number 4.*1425

*What about the length of time students used to complete a final exam within a 50 minute class period?*1428

*Let us put minutes on our X axis and the frequency over here on the Y axis.*1434

*Now since it is a 50 minute limit, 50 is going to be the max value people are should not be allowed to use 51 or 52 minutes.*1448

*The numbers are probably bunched up against that wall.*1460

*Remember skewed distributions usually happen when there some sort of imaginary wall in this case.*1466

*Probably most students might take a little less time, a little more time.*1473

*Maybe somewhere close to 50 and maybe some students will take all the way up to 50 minutes.*1480

*Maybe the students will be clustered around there and probably very, very few students will finish it in like 10 minutes or 20 minutes.*1488

*Maybe it will look something like this.*1499

*Fewer students are finishing it in 10 minutes but maybe there is one fast guy who does.*1513

*Maybe just a few more finishing it in 20 minutes but most of the students finishing around 40 or 50 minutes.*1518

*That is the last example problem, thanks for using www.educator.com.*1525

*Hi and welcome back to www.educator.com.*0000

*We are going to be talking about dot plots, and histograms in Excel today.*0003

*First we are going to talk about going from data to dot plots.*0009

*Remember before we always have to go from data to frequency tables and then to some visualization.*0013

*Dot plots are nice because they can let you go straight from data directly to the visualization.*0019

*We are going to talk about going from data to histograms.*0024

*Histograms are going to be really helpful to us especially because a lot of times we are going to have variables that have many values.*0028

*We are going to talk about how do we have grouped, ungrouped values, which we have looked up before, and grouped values.*0039

*Finally we are just going to talk a little bit about plotting frequencies versus relative frequency.*0046

*Relative frequency is just a fancy way of saying it is frequency but divided by how many cases you have.*0052

*It is really like percentage.*0060

*Previously we always have to go from data and we had to stop over at making a frequency table and then go to the visualization.*0065

*But now with dot plots we cut out the middleman and we can go directly from data to the visualization.*0073

*That is a really handy thing there.*0079

*If you look in your Excel file the data is going to be the same as the data we have been working with, the 100 www.facebook.com friends.*0085

*Here is what we are going to do, before we had created a nice little graph *0094

*using the Excel tools to create visualization but now we are going to use dot plots.*0100

*Excel will create dot plots for you directly.*0108

*Instead we have to sort of fidget, the fudge actually comes in handy sometimes.*0111

*Let us go birth month.*0119

*We already know that birth month should have a uniform distribution.*0121

*We already looked at this data before.*0125

*What we are going to do is just look at how to transform it directly *0127

*from data into a visualization without having to use the Excel graphs or chart.*0131

*If you go to your birth month sheet here I have just put up the months, 1 through 12.*0140

*It just looks sort of like a frequency table but if you watch carefully we are going to transform it.*0148

*Let us go ahead and put in our regular formula for how to find frequency.*0156

*That is equal sign (=) because we are starting off with a function.*0163

*Count if because we wanted to count if that person was born in the month of January.*0167

*Let us put in our data.*0177

*If we click on data and we scroll down to months, here is birth month.*0180

*I'm going to select all of these rows.*0188

*So far it seems like we are just making a frequency table.*0192

*I’m going to put in a comma because I know I’m going to need that.*0197

*Let us go back to birth month.*0201

*I want you to count it if the birth month is January.*0203

*If we just hit enter here, that would mean we are just counting how many people are born in January.*0210

*We are going to do something a little bit different.*0217

*I want Excel to visualize for me how many people there are.*0219

*Not give me a number but actually show me a pictures.*0224

*Here is what we are going to use.*0229

*We are going to use the repetition function and so that is rept, you do not have to put it in capital.*0230

*I just wanted to distinguish it from the count if and put a parentheses because that is how you are going to put in the inputs.*0238

*Here Excel reminds us that we need text, whatever text you want to repeat over and over again and the number of times.*0247

*The text, you can pick your favorite text.*0256

*You just have to make sure that it is in quotation marks.*0259

*I'm going to put in an at symbol, that is my favorite one.*0263

*I’m going to close parentheses and put in a comma.*0271

*The beautiful thing about count if is that it is going to return to me a number.*0274

*The output is going to be a number.*0279

*If I just leave this here it will just output to me 7.*0282

*This function will actually read repeat this at symbol 7 times.*0287

*At the end of this I’m going to put a close parentheses.*0296

*That my parentheses match up.*0302

*And then I’m going to hit enter.*0305

*Great.*0308

*Let me just make these rows a little bit larger so you can see everything.*0311

*Here instead of having the number 7, I have 7 little symbols.*0317

*And you do not have to use the at sign (@) if you do not want to.*0323

*You could use a star, an asterisk.*0327

*You could use anything you want.*0332

*You could use an o, anything to help you see there is this many people born in the month of January.*0333

*This is a direct way of going into the visualization.*0347

*We can actually copy and paste this just like we did before.*0351

*All we have to do is make sure that our data is locked with our dollar signs.*0357

*I’m going to hit enter and all I’m going to do is take that, drag it all the way down and we have a visualization right there.*0367

*We do not have to make Excel do any extra work with the charts or anything like that.*0377

*That is the handy thing about a dot plot.*0382

*We are making dot plot in Excel but a lot of times you may be asked to make dot plots on paper and pen.*0385

*No actual statistician does that anymore but in a statistics class you maybe asked to do that.*0392

*The nice thing about dot plots is that you do not have to put the data in order or anything.*0399

*You could just go and just sort of go down the line and put a dot where 5 is, *0404

*put a dot where 3 is, put a dot where 6 is, put a dot where 9 is.*0412

*You can see that it is a really easy way to visualize the distribution and you do not have to do anything to your data before hand.*0416

*But once again doing it by hand pretty tedious an*