For more information, please see full course syllabus of Statistics

For more information, please see full course syllabus of Statistics

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## Table of Contents

## Transcription

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### Introduction to Hypothesis Testing

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Roadmap
- Issues to Overcome in Inferential Statistics
- Issues to Overcome in Inferential Statistics
- What Happens When We Don't Know What the Population Looks Like?
- How Do We Know whether a sample is Sufficiently Unlikely
- Hypothesizing a Population
- Hypotheses
- Errors in Hypothesis Testing
- Steps of Hypothesis Testing
- Single Sample HT ( When Sigma Available)
- Single Sample HT (When Sigma Not Available)
- Example: Average Facebook Friends
- Step1: Hypothesis Testing
- Step 2: Significance Level
- Step 3: Decision Stage
- Step 4: Sample
- Sigma and p-value
- Example 1: Hypothesis Testing
- Example 2: Heights of Women in the US
- Example 3: Select the Best Way to Complete This Sentence

- Intro 0:00
- Roadmap 0:06
- Roadmap
- Issues to Overcome in Inferential Statistics 1:35
- Issues to Overcome in Inferential Statistics
- What Happens When We Don't Know What the Population Looks Like?
- How Do We Know whether a sample is Sufficiently Unlikely
- Hypothesizing a Population 6:44
- Hypothesizing a Population
- Null Hypothesis
- Alternative Hypothesis
- Hypotheses 11:58
- Hypotheses
- Errors in Hypothesis Testing 14:22
- Errors in Hypothesis Testing
- Steps of Hypothesis Testing 21:15
- Steps of Hypothesis Testing
- Single Sample HT ( When Sigma Available) 26:08
- Example: Average Facebook Friends
- Step1
- Step 2
- Step 3
- Step 4
- Single Sample HT (When Sigma Not Available) 36:33
- Example: Average Facebook Friends
- Step1: Hypothesis Testing
- Step 2: Significance Level
- Step 3: Decision Stage
- Step 4: Sample
- Sigma and p-value 45:04
- Sigma and p-value
- On tailed vs. Two Tailed Hypotheses
- Example 1: Hypothesis Testing 48:37
- Example 2: Heights of Women in the US 57:43
- Example 3: Select the Best Way to Complete This Sentence 1:03:23

### General Statistics Online Course

### Transcription: Introduction to Hypothesis Testing

*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

0 answers

Post by Thuy Nguyen on December 2, 2016

Hi Professor Son, I thought we reject the null when it falls below our critical value. But 7.97 is greater than 2.23.

Why did we reject the null?

0 answers

Post by Thuy Nguyen on December 2, 2016

Hello Professor Son, I don't understand why we didn't use the two-tail hypothesis test on the Example #1 (freezing water test). When and why do we use the one-tail hypothesis test vs. the two-tail hypothesis test?

1 answer

Last reply by: Professor Son

Tue Oct 28, 2014 1:06 PM

Post by Temitayo Akinshilo on October 26, 2014

When doing the SDoM drawings I see that you switch a lot from percentage and decimal format, it gets confusing. Also I spent a lot on the book is it possible to see you use the t and/ or z table from it as opposed to excel.

Thanks

0 answers

Post by Christopher Hu on December 25, 2013

Good stuff

0 answers

Post by Jennifer DeMott on March 16, 2013

Love the excel stuff!!! Keep it in!!!! It has really helped me learn how to use excel and how to do calculations way faster. However, notes are one reason why I might not continue service; they are so time consuming to download individually(why not in one PDF?)not to mention they have so many repeating pages with pretty much same info on them and then after printing 32 pages of notes, all the slides are not there. No example three this time!!!! Great lectures by the way!

0 answers

Post by Najam ul hassan Awan on January 4, 2013

Way way way too much dependence on excel!

Made me abuse her very badly !!!

0 answers

Post by Charles Forth on May 31, 2012

How do you calculate the p-value for the t- statistic without excel?