For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

### Sampling from a Normal Distribution

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
- Setting
- Assumptions and Notation
- The Sample Mean
- Standard Normal Distribution
- Converting to Standard Normal
- Example I: Heights of Students
- Example II: What Happens to This Probability as n → ∞
- Example III: Units at a University
- Example IV: Probability of Sample Mean
- Example V: How Many Samples Should We Take?

- Intro 0:00
- Setting 0:36
- Setting
- Assumptions and Notation 2:18
- Assumption Forever
- Assumption for this Lecture Only
- Notation
- The Sample Mean 4:15
- Statistic We'll Study the Sample Mean
- Theorem
- Standard Normal Distribution 7:03
- Standard Normal Distribution
- Converting to Standard Normal 10:11
- Recall
- Corollary to Theorem
- Example I: Heights of Students 13:18
- Example II: What Happens to This Probability as n → ∞ 22:36
- Example III: Units at a University 32:24
- Example IV: Probability of Sample Mean 40:53
- Example V: How Many Samples Should We Take? 48:34

### Introduction to Probability Online Course

### Transcription: Sampling from a Normal Distribution

*Hello, welcome back to the probability lectures here on www.educator.com, my name is Will Murray.*0000

*Today, we are going to talk about sampling from a normal distribution, which is really starting to get into statistics.*0006

*Sometimes it is still considered as a topic in probability.*0013

*We are going to go ahead and talk about it.*0017

*We are not using the Central Limit Theorem,*0020

*I have another lecture on the Central Limit Theorem that is going to come after this one.*0022

*If you are looking for Central Limit Theorem, just skip ahead and look for that video.*0026

*In the meantime, we are going to learn how to take samples from a normal distribution*0030

*without using the Central Limit Theorem.*0035

*Let us see what this is all about.*0037

*First of all, I have to tell you what it means to take samples.*0040

*The idea is that, we have some kind of population of stuff and that can be almost anything.*0042

*The example I have here is, we have, maybe students in the university and the students are all different heights.*0048

*We are going to select some students at random and measure their heights.*0054

*That is taking a bunch of samples, it could be that you are testing, *0060

*for example, a soda machine and you are testing whether it dispenses the right amount of soda.*0064

*You fill up 10 cups of soda and you measure how much they put in each cup.*0070

*That is the same kind of idea, you are taking a bunch of samples of the population.*0075

*The population that we are studying has a mean ν and the variance σ².*0080

*We do not always know those, in this lecture we are going to need to know the variance but we will not always know the mean.*0085

*A lot of examples that we will be studying, we would not know the mean of the population.*0091

*As I mentioned, we are going to take samples.*0098

*If we are talking about students and different heights, then our Y1 would represent the first student that we sample.*0100

*We meet a student at random and then we measure how tall that person is.*0106

*We meet another student at random and measure how tall that person is, and that is Y2.*0110

*We meet another student at random, and so on, until we meet our last student which is the YN.*0115

*We measure the height of that last sample student.*0123

*That is what it means to take samples, you have a population, *0126

*you select some of them at random and you measure whatever quantity you are interested in*0129

*for each one of those of random selections.*0135

*Let me show you some assumptions that we need to get started.*0138

*The assumption that we are going to make for this lecture and for the next one, is that our samples are independent.*0142

*If we happen to find a couple of tall students, it does not make it more or less likely*0149

*that the student after that will be extra tall or extra short.*0154

*The samples are totally independent of each other, that let us use some of our theorems about random variables.*0157

*There is a buzz phrase that people use in probability and statistics, it is independent identically distributed.*0163

*That is often shortened to IID because that phrase is used so often.*0172

*IID means independent identically distributed random variables.*0177

*The independent part is the assumption we just made.*0182

*Identically distributed means that they are coming from the same population.*0185

*Taking students from the same university or we are measuring how much a soda machine dispenses *0189

*and we are taking samples from the same soda machine, that kind of thing.*0197

*The assumption for this lecture only, for this lecture on normal population, is that our population has a normal distribution.*0201

*That is the key difference between this lecture and the next lecture.*0211

*The next lecture is on the central limit theorem.*0214

*Up till now, everything else is the same.*0216

*In the next lecture, we would not need to assume their population is normally distributed.*0218

*In this lecture, we are assuming normal distribution because we are not using the central limit theorem.*0224

*The notation that we use for that normal distribution is that, *0230

*we say each of our samples has a normal distribution and it has mean ν and variance σ².*0234

*That is the notation that we use for a normal distribution.*0243

*The first variable is always the mean and the second variable is always the variance there.*0249

*What we are going to do is take all of our samples, we will measure the heights of each student.*0257

*And then, we will take the average of the heights that we measure, that is called the sample mean.*0263

*We often use Y ̅ as a notation for the sample mean.*0268

*It just means the average, it just means we take all the qualities that you measure*0271

*and you add them up, and you divide by N.*0275

*The idea is, we are going to use the sample mean and ask questions about how it relates to the actual mean of the population.*0280

*It is not necessarily true that the average of the students that I study is equal to the average of the entire student population.*0289

*The examples in this lecture are, all have to do with questions about how close are those 2 means to each other.*0298

*What is the mean of the students we study, that is the sample mean.*0305

*What is the mean of the entire population?*0309

*That was the population mean which is not the same as the sample mean.*0312

*The population mean, remember we were saying was μ.*0319

*That is what we mean by the population mean, it is the average of all the students at the entire university.*0324

*The sample mean is just the mean of the few students that we select to study and actually measure 1 by 1.*0331

*The theorem that we are going to be using here is, the assumption we already made is that*0341

*each of one of the variables Yi has a normal distribution with mean ν and variance σ².*0347

*Then, the sample mean also has a normal distribution.*0355

*It also has mean ν but its variance is σ²/N.*0359

*It actually has a smaller variance than the individual random variables, representing the individual measurements.*0366

*Just to summarize that words, Y ̅ the sample mean is a normally distributed random variable*0377

*with mean ν and variance σ²/N.*0383

*This is the key to the whole idea of sampling which is that, as you take more samples that means N grows.*0387

*N is the number of samples that you take.*0396

*If you take more samples, it shrinks the variance of your mean, *0397

*what it means that your sample mean is more likely to be accurate.*0403

*It is less variable, that is why it is more accurate to take a survey with a large number of people*0409

*than with a small number because it shrinks the variance.*0415

*This is the mathematical principle that guarantees that.*0419

*What we are going to do is, we are going to use the fact that *0426

*we have a normal distribution on this sample mean to ask questions about probabilities.*0430

*The whole point of asking questions about probabilities is that,*0440

*we can convert normal distributions to standard normal distributions.*0445

*I will show you the equation for that on the next slide.*0449

*Let me just mention that we can answer questions about standard normal distributions using charts.*0452

*That is what I got on this slide, right here, a standard normal distribution. *0459

*We often use the variable Z for a standard normal distribution.*0463

*What that means is that, it is a normal distribution with mean 0 and variance 1 or standard deviation 1.*0467

*That is what the standard normal distribution means.*0477

*We got mean 0 and variance 1.*0481

*The whole point of standard normal distributions is that, there are standard charts that you can look up probabilities on.*0486

*We will be using this to solve the problems later on in the lecture.*0492

*What all these little numbers represent on the chart is*0496

*the probability of being above a certain cut off for a standard normal distribution.*0499

*For example, if you want to say what is the probability that Z is greater than 0.93, for example.*0506

*I would look at 0.9 here and I see the second decimal place is 0.03.*0520

*I see that, that number on my chart there is 0.1762.*0527

*My answer here would be 0.1762.*0534

*That is how we look up to a probability for a standard normal distribution.*0539

*It is also frequently asked, what is the probability that Z is less than something or between to bounds.*0546

*What you have to do is always keep this picture in mind.*0555

*If you want to find the probability that Z is less than something, *0560

*what you do is you use this chart to find the probability that Z is greater than that cutoff, then do 1 - that probability.*0564

*Let me write that down, the probability that Z is less than some cutoff is equal to 1 – *0573

*the probability that Z is greater than some cutoff.*0580

*We will see some examples of that in the exercises.*0584

*You can also talk about the probability of Z being between 2 cutoffs.*0588

*You can also flip these numbers around to get probabilities for negative values of Z.*0592

*We will practice all of that in the examples.*0599

*Remember, this is just for a standard normal distribution, when we have mean 0 and standard deviation 1.*0602

*Let me show you now what you do for a nonstandard normal distribution.*0609

*If you have a nonstandard normal distribution, what you do is you take any normal distribution.*0614

*To make it standard normal, you subtract off its mean and you divide by its standard deviation.*0623

*And then, what you get we are going to call that Z is a normal distribution that is standard.*0630

*The mean is 0 and the standard deviation is 1.*0638

*Let me remind you of that theorem we had.*0642

*We had that theorem that said that Y ̅ is a normal distribution.*0644

*Its mean is μ and its variance is σ²/N.*0649

*That means its mean is μ and the standard deviation, remember is always the square root of the variance.*0657

*Standard deviation is the square root of σ²/N, which is σ/√N.*0663

*What we do to convert to Y ̅ to a standard normal is, we do Y ̅ - its mean.*0676

*Y ̅ - μ divided by its standard deviation, we do σ divided by √N.*0683

*Since that is in the denominator, a fraction, I’m going to do a little flip.*0692

*I will get our √N × Y ̅ - μ divided by σ.*0698

*That, according to my theorem is a standard normal distribution.*0708

*If you call that Z then that is a standard normal distribution.*0713

*It means I can use those charts, the chart that I showed you on the previous side, to look up probabilities for Z.*0718

*That is the way you play this game.*0725

*Often, the way the examples pen out is you are asked something about Y ̅.*0727

*Often, you are asked about the relationship between Y ̅ and μ.*0733

*How likely is it that Y ̅ and μ are within 1 unit of each other, something like this.*0737

*You are trying to solve something about Y ̅ – μ.*0743

*The trick here is to take Y ̅ – μ, and you are asked how likely is that Y ̅ - μ is within ½ unit of each other, something like that.*0747

*The trick to solving these things is to convert it to a standard normal.*0760

*The way you convert to standard normal is you multiply by √N/σ.*0764

*That is a standard normal variable and you can look up probabilities for that on the charts.*0773

*We will get a lot of practice of that in the exercises, but I want you to see the general idea first,*0780

*which is that you start with Y ̅ - μ and then you multiply by √N/σ to convert it into a standard normal variable.*0787

*I will go ahead and practice some exercises with that.*0796

*The first example, we are going to measure the heights of students at a particular university.*0800

*We are given that they are normally distributed, that means we can use the theorems*0805

*that we have learned from this lecture.*0809

*There is a standard deviation of 4 inches, that is going to be the σ that we are going to use later on.*0812

*We are going to measure the heights of 9 students.*0819

*We are going to sample 9 students, we are just going to go out in the crowd at the university *0822

*and grab the first 9 students that we see.*0826

*We are going to measure how tall they are.*0829

*The question is, what is the probability that their mean, *0831

*that means the mean of the students that we study is within 2 inches of the global means.*0835

*That is the mean of all the students at the university.*0841

*I will just clarify the difference between those 2 means here.*0846

*When I say the global mean, that is all students at the university, *0849

*all the students at enormous state university that you attend, or maybe you do not.*0855

*Their mean, when I say their mean that means the sample mean of the students that we include in our sample.*0864

*That is those 9 students that we are going to study.*0871

*The global mean, the notation we have been using for that is μ, the sample mean is Y ̅.*0875

*The question is, what is the probability that those 2 means are within 2 inches of each other?*0884

*Within 2 inches of each other, that means + or -2 inches in either direction.*0891

*We are trying to solve the question of Y ̅ – μ, the sample mean - the global mean.*0896

*We are going to put absolute values to get myself within 2 inches.*0906

*The absolute value of A - B means the distance from A to B.*0910

*What is the probability that the distance from the sample mean to the global mean is less than or equal to 2?*0915

*That is what we are trying to solve, we are trying to find the probability of that.*0923

*But remember, the trick here is to convert to a standard normal distribution.*0927

*The way you convert to a standard normal distribution is, you multiply both sides Y ̅ – μ.*0933

*We multiply both sides by √N/σ.*0942

*I will do the same thing on the right, 2 √N/σ.*0953

*Let me fill in now what I know, I know that √N, N = 9, that is the number of students that we surveyed here.*0960

*√N would then be 3, that is 2 × 3.*0972

*The σ is the standard deviation, that is 4 inches.*0982

*2 × 3/4, that is a very easy thing to simplify, that is just 3/2, that is 1.5.*0992

*The point of that is that, that was a standard normal variable.*1001

*We are asking about the absolute value of Z being less than or equal to 1.5.*1005

*We are trying to find the probability that a standard normal variable will be less than or equal to 1.5.*1012

*Let me draw a little graph here and show you what kind of area we are looking for.*1019

*We will actually look it up on the next slide.*1024

*What we are looking for here, I will draw my standard normal variable centered at 0 because it has mean 0.*1027

*That is supposed to be symmetric.*1034

*Here is -1.5 and I'm looking for that probability in between those two bounds right there.*1036

*That is not exactly what the chart will tell me directly.*1048

*What the chart will tell me is, if I have a particular Z value in mind, remember what the chart tells me.*1052

*It will tell me the probability that Z is bigger than that value.*1060

*I have to figure out from that, what my probability is that Z is between -1.5 and 1.5.*1065

*The probability that the absolute value of Z is less than or equal 1.5.*1073

*I see I can get it by taking that outside probability and subtracting off two copies of it, because there is a bottom tail and a top tail there.*1080

*It is 1 -2 × the probability that Z is greater than 1.5.*1089

*That is something that I will be able to look up on my chart, on the next slide.*1096

*I just wanted to make sure that you understand where it is going to come from.*1100

*I will look that up and that will give me my answer to this example.*1104

*Let me recap the steps here, before I turn the page.*1109

*I'm setting up a sample mean Y ̅ and a global mean μ.*1113

*I want them to be within 2 inches of each other.*1120

*That means, I want their difference to be less than 2 in absolute value.*1124

*And then, I'm kind of building up my standard normal variable by multiplying that by √N/σ.*1128

*Remember, the whole point was √N/σ × Y ̅ - μ is the standard normal variable.*1135

*I’m building up that expression here, I multiply by √N/σ.*1150

*My N was 9, √N was 3, my σ was 4 because that was the standard deviation given to me.*1155

*I can simplify that down to 1.5, I want the probability that a standard normal variable*1163

*will be less than 1.5 and absolute value, which really means between -1.5 and 1.5.*1168

*Which means to calculate that area, I am going to have to look at the tail and subtract off two copies of the tail,*1176

*because there is a top tail and bottom tail there.*1183

*That is why I'm going to multiply this probability by 2, the probability that I will figure out on the next slide.*1186

*Let me go ahead and figure out that probability using the chart.*1193

*What we figured out is that, our probability that Z is less than or equal to 1.5.*1199

*This is coming from the previous slide, you can scroll back and watch it if you like.*1208

*That is 1 -2 × the probability that Z is greater than or equal to 1.5.*1212

*I need to find 1.5 on my chart, there it is right there 1.5, it is 1.50.*1220

*I’m going to take this number right here, 0.0668.*1228

*1 - 2 × 0.0668, and that is 1 – 0.668 × 2 is 1336.*1234

*It is 1 -0.1336 and 1 -0.1336 is 0.8664.*1257

*If you wanted to estimate that as a percentage then that is about 87%.*1271

*That is the answer to my problem.*1277

*If I survey these 9 students and measure their heights,*1281

*there is an 87% chance that my sample mean will be within 2 inches of the global mean of the population.*1285

*That is how likely I am to get an accurate estimate, when I survey 9 students.*1294

*If I want to make it more accurate, I will survey more students because that would increase the value of N in my calculations.*1302

*Just to recap what we did on this side, we figured out on the previous side, we want Z to be less than 1.5 and absolute value.*1310

*We figured out that, to get Z less than 1.5, what we can do is look at these two tails *1320

*and cut off the two tails that represent the probability of Z being bigger than 1.5.*1327

*That is why I have 1 -2 × the probability of Z being bigger than 1.5.*1335

*Then I found 1.50 on my chart, 1.50 and there it is 0.0668.*1340

*Plug in that number, do a little calculation and got to my answer of 87%.*1349

*In our second example here, we re given that Y1 through YN *1358

*are independent identically distributed random variables, that is what that phrase IID means.*1363

*It means independent identically distributed.*1371

*It comes up so often in probability that people just use that abbreviation for it.*1373

*Each Y, it has a normal distribution with mean μ and variance σ².*1378

*We are told that σ² is 64 and N is 36.*1386

*N is the number of samples that we are going to be taking.*1390

*We want to find the probability that the sample mean Y ̅ is within 1 unit of the global mean μ.*1393

*We are also asked, what happens to this probability as N goes to infinity?*1401

*Let us think about that, remember Y ̅ is the sample mean.*1408

*It is the average of the samples and we want that to be close to μ which means we want Y ̅ - μ to be small.*1412

*We want it to be less than 1.*1421

*What I would like to do is build up a standard normal variables so I can use my theorem here.*1428

*I'm going to multiply just like before, by √N/σ.*1436

*I will √N/σ on the right hand side, I got to do the same thing to the left and right hand side.*1444

*The point of that was that, that will give me a standard normal variable.*1451

*I’m putting absolute values on the Z because I have the absolute values on Y – μ there.*1456

*In this case, we are given that N is 36, √N = 6.*1460

*Let me say that Z is less than or equal to 1 × √N is 6, σ² is 64 and that tells me that σ would be 8.*1469

*We are given the variance, instead of the standard deviation that time.*1483

*It is a quick maneuver to go from the variance to the standard deviation.*1487

*N the standard deviation is always the square root of the variance.*1491

*In this case, we got 8 and that is ¾.*1493

*I need to find the probability that a standard normal variable is going to be less than or equal to ¾.*1498

*Let me draw a little picture here.*1510

*There is ¾, we want it to be between ¾ and - ¾.*1518

*We are looking for that area right there.*1523

*Just like in example 1, the way we can calculate that area is by finding the area outside there, the tail area.*1526

*And then, subtracting off two copies of the tail.*1533

*The probability that Z is less than or equal to ¾ is equal to 1 - 2 × the tail area, 2 × the probability that Z is bigger than ¾.*1539

*I got a standard normal distribution setup on the next slide.*1561

*But I think what I'm going to do is just tell you what the answer is right now, *1567

*I can go ahead and use that space on this slide to show you the rest of the calculations.*1571

*I will justify this number on the next slide, but what we are going to find is we will look up 0.75 on the next slide.*1576

*We will figure out that the probability, the tail area of being bigger than that is 0.2266.*1584

*That is what I looked up on the next slide.*1595

*This is 1 -2 × 0.2266, and in turn that is 1 -, 2 × that is 0.4532.*1599

*1 - 0.4532 is 0.5468 or approximately 55% there.*1613

*That is the probability that you are going to be within 1 unit of the global mean μ, when you take these samples.*1623

*The second part of this question here says, what happens to this probability as N goes to infinity.*1635

*Let us think about it, as N goes to infinity that means your N is getting bigger and bigger.*1641

*Which, if we trace through these calculations, that means that N would get very big.*1647

*That in turn, would make that number very big, get bigger and bigger.*1655

*That number is big, which means what you are doing is you are kind of moving these goal posts farther and farther out.*1660

*This ¾ would get replaced by a bigger number.*1674

*You would be looking at a wider range of your normal distribution.*1680

*That probability would get bigger and bigger.*1686

*Another way to think about it is that, the probability of being in the tail would get smaller and smaller.*1690

*The probability of being in the tail would get smaller.*1696

*The overall probability would get bigger and bigger.*1702

*In fact, as N goes to infinity that probability would go to 1*1709

*because you encompass more and more of the area, showing that you have more and more probability.*1714

*The probability goes to 1, as N goes to infinity.*1720

*That should make sense to you, it is kind of the precise version of saying that as you take more samples,*1724

*your probability of being accurate is higher and higher and in fact, it approaches 1.*1733

*If you take infinitely many samples then you are guaranteed to get an accurate average.*1738

*Let me recap the steps here, before I jump onto the next slide and show you where that 1 number comes from.*1745

*I still owe you that 0.2266, that part is still mysterious.*1751

*I started out with Y - μ being less than or equal to 1.*1755

*That came from this phrase here, Y is within 1 unit or Y ̅ is within 1 unit of μ.*1760

*I want to build up a standard normal variable.*1769

*I multiply by the √N/σ.*1772

*I multiply both sides there by √N/σ.*1775

*The N was 36 which means the √N is 6, that is √N right there.*1778

*I was told σ² was 64, that tells me σ is 8, that is where that 8 comes from.*1786

*6/8 collapses to 3/4 which means I'm looking at the region between - ¾ and ¾.*1796

*The clever way to calculate that using the chart is, to find the tail region bigger than ¾ *1805

*because that is what our chart will do for us.*1811

*The probability of being in that tail region, this is the part I’m going to look up on the chart on the next slide, is 0.2266.*1814

*That is the only part that should not make sense yet, until you see the next slide.*1822

*And then, when I plugged in 0.2266 and I simplified the calculations, *1826

*it just reduced down 2.5468 or just about 55% is my probability.*1831

*I went through and trace the role of N, in those calculations.*1838

*I noticed that, if you put in a bigger and bigger N, we get a bigger cutoff for the bounds of the region here.*1842

*When you extend the bounds outwards, it means your tails are getting smaller.*1849

*The tails is what we subtracted, the total probabilities can get bigger and bigger, and go to 1 as N goes to infinity.*1855

*That should conform with your intuition because it means that, as you take more and more samples,*1865

*you are more and more likely to get accurate estimations of the global mean.*1873

*That is sort of reassuring that it worked out that way.*1878

*Let me show you where this number comes from, this 0.2266, that is the one part that I have not shown you yet.*1881

*That comes from this chart right here of the normal table, we are trying to find the probability that Z was greater than ¾.*1887

*Of course, ¾ in decimal is 0.75.*1899

*I just have to find 0.75 on this chart, here 0.7 over here and here is the 0.05.*1904

*I find their intersection, there it is 0.2266, that is the answer that I plugged into my calculations on the previous slide.*1915

*That was the only missing piece of the puzzle on the previous slide.*1930

*Everything else is supposed to make sense and you are supposed to understand the answer to example 2 now.*1934

*In example 3, we have students at a university and we are going to keep track of the number of units that they have taken.*1946

*It turns out that they have taken an average of 70 units, but their standard deviation is 20 units.*1954

*Most students are probably somewhere between 50 units and 90 units, averaging around 70 units.*1960

*We are going to assume that this is a normal distribution.*1966

*We are going to sample 9 students.*1970

*We are just going to go out and meet 9 students in the quad.*1972

*We are going to say, how many units have you taken?*1977

*We will do that 9 × and then we will calculate the average, just of those 9 students that we have met.*1981

*We do not have the time to meet every student in the university, *1989

*we will just sample 9 of them and calculate the average unit load of those 9 students.*1992

*The question is, are we likely to get an average between 67 and 73 units?*1997

*Let me show you how you think about that.*2005

*67 is 3 units down from the mean, that is 70 -3.*2008

*73 is 3 units up from the mean.*2015

*What we are really asking is, what is our chance that we will be within 3 units of the global mean.*2019

*The global mean is 70 units, the mean of the students that we are surveying is Y ̅.*2027

*We want Y ̅ - μ here to be 3 units, to be less than or equal to 3 units.*2034

*That is to get Y ̅ between 67 and 73, that is what we want to study there.*2043

*Remember, the whole point is that we want to convert this into a standard normal variable.*2054

*Our standard normal variable is always √N/σ × Y ̅ – μ. *2060

*I'm going to multiply on some factors of √N/σ.*2068

*√N/σ × Y ̅ - μ is less than or equal to 3 √N/σ.*2075

*The whole point of that was that gives me a standard normal variable.*2086

*I want to find now the probability that a standard normal variable will be within this range.*2092

*√N, what is my N, that is the number of students that I'm sampling, that is 9.*2098

*√N is going to be 3, that is coming from the √9 there, that is not the 3 that I found up above.*2103

*The σ is the standard deviation of the population which is given to me to be 20 units.*2113

*That is going to be 20 and my Z should be less than or equal to 3 × 3/20.*2123

*9/20, I can convert that into a decimal, I think it is going to be useful 1/20 is 0.0579.*2134

*9 of those is 0.45.*2141

*We really want to find that probability that Z and absolute value is less than 0.45.*2147

*Let me draw a picture of what I'm trying to calculate here.*2155

*It is always very useful to draw pictures of these normal distributions.*2158

*I hope to keep track of what you are looking up on the table.*2162

*In this case, I wanted to be between -0.45 and 0.45.*2165

*Those were not very symmetric there, that should be symmetric because 0.45 is the same distance on either side, there is 0.45.*2172

*I'm looking for that area in between there.*2182

*The probability that the absolute value of Z is less than 0.45.*2186

*Another way to find that would be the probability that -0.45 is less than Z, is less than 0.45.*2196

*That is not something that the table will tell me directly.*2206

*Remember, the table will tell me how big, how much area I have in the tail of the distribution.*2208

*What I will do is, I will find the area and the tail from the table.*2215

*It looks like that I have to subtract 2 tails there.*2220

*1 - 2 × the probability of Z being bigger than 0.45, that is what I'm going to look up on the next slide*2223

*and actually convert that into an answer.*2234

*Let me just to go over the steps again quickly for this slide.*2238

*I was given that I'm looking for unit total between 67 and 73.*2241

*If I want Y ̅ to be between 67 and 73, that is + or -3 from the mean of 70.*2246

*It is the same as saying Y ̅ - μ is less than 3.*2255

*I wanted to convert that into a standard normal variable.*2259

*I multiplied both sides by √N/Σ, in order to build up my standard normal formula.*2263

*My Z is now less than or equal to, 3 × √N.*2271

*I was given that N = 9, where did that come from?*2277

*That is number of students that you sample.*2279

*My standard deviation is 20, that is my σ right there and the 9 was the N.*2283

*I plug those values in and I get up absolute value of Z is less than 0.45.*2292

*And then, I did a quick little picture to see what kind of area I’m measuring.*2297

*I see that the way to measure that area is really to measure the tails, and then subtract the 2 tails from 1.*2303

*That is what I'm going to carry over onto my neck slide and solve it out using a normal chart.*2310

*This is kind of the rest of the example 3, we are going to use the normal chart.*2317

*We figured out that, we are looking for the probability that Z is less than 0.45.*2321

*The absolute value of Z is less than 0.45 which is 1 - 2 × the probability that Z is greater than 0.45.*2326

*Remember, that is what we are finding with this normal chart.*2340

*It will tell you the amount of area in the tail there, the probability in the tail.*2342

*I need to find 0.45 on this chart.*2347

*Here is 0.4, here 0.5, I see 0.3264 at the intersection of that row and column.*2350

*1 -2 × 0.3264 and 2 × 0.3264 is 0.65, 64 × 2 is 128, .028.*2359

*That is 0.3472 or approximately 35%, that is the probability that the 9 students*2388

*that I survey will have their average unit load somewhere between 67 and 72.*2404

*What I really calculated there, in other words the steps in the middle, was the probability that Y ̅ is between 67 and 73.*2412

*Most of this was done on the previous page.*2426

*Most of the dirty works was done on the previous page.*2427

*All that I did on this page was, I use the chart to find the probability that Z was less than 0.45.*2429

*In order to figure that out, I subtracted off 2 tails here.*2438

*I looked up the value of the area in that tail, and then I just did *2442

*a little simplification with the numbers and reduced it down to 35%.*2447

*An example 4, we are going to take 6 samples from a normally distributed population with variance 0.67.*2455

*We want to find the probability that the sample mean, *2463

*the average of our samples will be within 0.5 units of the population mean.*2465

*Let us calculate that out.*2471

*The sample mean is Y ̅, that is the average of the samples that you have taken.*2474

*The population mean is always μ.*2481

*Even though, you do not know exactly what the value of μ is, we always call the population mean μ.*2484

*The probability that they will be within 0.5 units of each other.*2490

*We want Y ̅ - μ to be less than 0.5 in absolute value.*2495

*Let me build up my standard normal variable, as usual.*2503

*I will multiply this by √N/σ × Y ̅ – μ.*2508

*That should be less than or equal 2.5 × √N/σ.*2517

*I actually rigged the numbers for this one, it is supposed to work out fairly well.*2522

*Let me show you how it works out.*2526

*My N was 6, there is N = 6 there.*2528

*I rigged this 0.67 to be equal or very close to 2/3.*2532

*I hope it actually turns out to work.*2540

*That was the variance, that was σ².*2543

*What we have here is a0.5 × √6 divided by, σ by itself will be √2/3.*2549

*The whole point of this was that, this was Z that is supposed to be a standard normal variable.*2561

*We want Z in absolute value to be less than or equal 2.5 ×, I can simplify that in 2.5 × √6 divided by 2/3.*2567

*If I do a little flip on the denominator, I will get 0.5 × 6 × 3/2.*2581

*That is 0.5 × √9 and that is 3/2 or 1.5.*2591

*What I'm really looking for is the probability that the absolute value of Z will be less than 1.5.*2599

*Since I know I have a chart that will tell me the tail, the area in the tail of the distribution.*2608

*That is what the chart will tell me, the area and the tail.*2615

*What I want is the probability that Z is less than 1.5, that means the absolute value of Z is less than 1.5.*2618

*Z is between -1.5 and 1.5.*2628

*The way I can figure that out is I can subtract off 2 tails.*2634

*1 -2 × the probability that Z is bigger than 1.5.*2639

*That is really all I need to do for now, I’m going to look at a standard normal chart on the next slide.*2646

*And, I will go ahead and finish that calculation.*2651

*Let me recap the steps here.*2654

*We want Y ̅ – μ.*2656

*Y ̅ is the sample mean and μ is the population mean.*2657

*Those mean different things.*2662

*The population mean means the entire population.*2664

*Sample mean means just the samples that we are looking at.*2666

*We want to be within 0.5 units of each other.*2671

*I said it is less than 0.5 and I cannot do much for that, until I convert it to a standard normal variable.*2676

*That is what I'm doing here, multiplying both sides by √N/σ.*2682

*I know what N is, it is √6.*2688

*I know what σ is, it is √2/3, it came from there.*2690

*I plug those in, it worked out fairly nicely and I got o3/2 or 1.5.*2699

*We are going to find the probability of being between -1.5 and 1.5.*2705

*They way that I’m going to do that is, by finding the probability of being in the tail and then, subtracting off 2 of those tails.*2710

*I will make that out in the next slide.*2720

*Let me just mention right now, before I bury this slide, that are we are going to use the same setup in example 5.*2722

*The only difference is we are going to change the number of samples, in order to get a better probability.*2731

*Make sure you understand this, before we move on to example 5,*2737

*because example 5 would not make sense, unless you understand example 4 here.*2741

*Let me just flip over to our chart of the normal distribution and we will finish this problem.*2748

*On the previous page, I had solved it down to the probability is 1 -2 × the probability that Z is bigger than 1.5.*2753

*I solved it out into a matter of finding the probability of the tail of the distribution.*2769

*I need to find 1.5, there it is 1.50, the probability is 0.0668.*2776

*This is 1 -2 × 0.0668 and just a little computation is all we have to do here.*2788

*That is 1 – 0.668 × 2 is 0.1336.*2799

*If I simplify that, 1 -0.1336 is 0.8664 and that is approximately 87%.*2813

*If you take 6 samples and you want to find the probability that Y ̅ - μ was less than 0.5, that is what we just calculated here.*2826

*We found the probability that our sample mean will be within 0.5 units of the global mean, it worked out to 87%.*2848

*Most of the work there was on the previous page.*2855

*I just kind of brought it down to looking at one number in the chart, and then just plug it in.*2858

*That one number was the probability that Z is bigger than 1.5.*2863

*We found that probability from the chart here, drop it into the computation, and reduced it down to 0.8664 or 87%.*2869

*We are going to reuse this scenario in example 5.*2878

*I really want to make sure the example 4 makes sense to you, before you go on to example 5.*2882

*In particular, what we are going to do in example 5 is, we are going to try to raise that probability to 95% in example 5.*2888

*Which means we are going to have to change the number of samples that we take.*2899

*In order to get higher probability and more accurate answer, we will need to take more samples.*2907

*Let us go ahead and take a look at example 5, and see how that works out.*2912

*In example 5, we are going to reuse this scenario from example 4.*2916

*If you have not just watched example 4, you really want to go back and watch example 4.*2921

*Make sure that that make sense to you, before you start working through example 5.*2925

*It is the same scenario as in example 4.*2932

*We have a normally distributed population with variance 0.67.*2935

*I want to ensure that our sample mean, our sample mean*2940

*means the average of the samples that we take, will be within 0.5 units of the population mean.*2943

*The population mean is the average of the entire population, that is what we have been calling μ.*2952

*We want the probability to come out to be 95%.*2958

*Since, we are dictating the probability, we cannot dictate the number of samples.*2962

*We are asking, how many samples should we take?*2965

*Let me work this out and show you how to think about this.*2969

*First of all, we want our probability to be 95%.*2975

*Let me think about that, in terms of the picture.*2979

*I will draw a picture there.*2984

*We want some cutoffs where we get 95% of the area in between those cutoff.*2985

*We want those cutoffs to surround 95% of the area.*2993

*I work backwards, that means that the 2 tail areas collectively give me 5% of the area.*2999

*Those 2 tail areas 1 -0.95/2 which is 0.05/2 which is 0.025, *3011

*I'm going to want the probability in the 2 tail areas to be a 0.025 each.*3023

*I want to figure out what cutoff value of Z would correspond to that.*3031

*I want to save myself space on this slide, I’m not going to show you the chart right away.*3037

*We will see that on the next slide.*3041

*You will see that that corresponds to Z = 1.96.*3043

*We will look that up on the next slide and you will see that that is the Z value we are looking for.*3047

*Let us put that on hold for now and let me go back and set up our standard normal variable.*3052

*We want Y ̅ – μ to be within 0.5 units of each other.*3060

*I’m going to set up my standard normal variable just like before, where I multiply both sides by √N/σ.*3067

*I get √N/σ here.*3077

*I will fill in what I can, the problem is I do not know N right now.*3081

*That is going to be a little tricky, my Z is going to be the standard normal variable on the left.*3086

*But, I do not know what N is.*3092

*I do know what I want my Z value to be, or at least I know that I want my Z value to be between -1.96 and 1.96.*3094

*I’m going to put 1.96 in for my Z value, my absolute value of Z.*3104

*And then, I'm going to solve for the other quantities in this picture.*3109

*0.5 √N, I do not know that, that is what I'm going to have to solve for.*3114

*Σ, I think I do know.*3119

*I'm given variance is 0.67, that tells me that σ² is the variance is 0.67, that is √0.67.*3122

*What I’m going to do is solve this equation for N.*3136

*It is going to work out pretty well, it is a calculator exercise really.*3143

*If I multiply over to the other side, I get 1.96 × √0.67 divided by 0.5 is less than or equal to √N.*3148

*You know what, dividing by 1/5 is the same as multiplying by 2.*3163

*Let me go ahead and multiply 1.96 by 2, that will give me 3.92.*3168

*I still have √0.67, and that is supposed to be less than √N.*3176

*Let me square both sides now.*3182

*I think I’m going to flip the N over to the other side.*3184

*N is bigger than or equal to 3.92².*3186

*If I square 0.67, the square root of that, I will just get 0.67 again.*3192

*Now, that is just a matter of dropping the numbers into a calculator.*3199

*I did that, when I drop that into a calculator I get 10.296.*3202

*I just solved for N, and remember that N is the number of samples we are going to take.*3216

*You cannot take a fraction of a sample, you take a whole number samples.*3223

*This 10.296 does not make sense, I'm going to round it up to be on the safe side.*3232

*I will take N = 11 samples and that should be enough to get my probability where I want it to be.*3240

*That is my answer right there, N = 11 samples.*3250

*That is the end of the problem, except to recap it and to show you that on a normal table where that 1.96 came from.*3255

*Let me recap the steps there.*3264

*First, I was thinking that I wanted to have 95% in between whatever boundaries I found,*3266

*which means on the outside of boundaries I’m going to have 5%, 0.05.*3272

*Since, there are two tails, I will divide that by 2 and I got 0.0252.*3277

*I’m looking for a cutoff that cuts off 0.025 of the area.*3283

*I will show you on the next slide that, when we look at the normal table, Z = 1.96 will give us that cut off.*3289

* That is the only part that I need to fill in on the next slide.*3300

* Meanwhile, over here I was setting up my standard normal variable.*3303

*I wanted Y ̅ - μ to be within 0.5 units of each other.*3307

*That is why I set their absolute value of their difference less than 0.5.*3314

*And then, to set up my standard normal variable, I multiplied top and bottom by √N/σ*3319

*and that gives me my absolute value of Z.*3328

*Now, I plugged in that Z = 1.96 here.*3329

*I do not have √N because I do not know what N is.*3333

*That is what I'm asking, how many sample should I take?*3338

*I have to leave that, but I can plug σ which I figure it out here to be √0.67.*3340

*Now, it is an algebra problem, I have manipulated the algebra a little bit.*3348

*1.96 divided by ½ is the same as multiplying by 2.*3352

*That is where that 3.92 came from.*3357

*I’m solving for N, I square both sides and I get N bigger than 3.92² × 0.67.*3360

*I just threw those numbers in my calculator, I would not want to do something like that by hand.*3368

*What I got was 10.296, but since we are talking about numbers of samples.*3374

*We have to take a whole number of samples.*3379

*I rounded that up to be safe to take, N =11 samples.*3381

*That pretty much wraps up example 5, except I have to show you where that 1.96 came from.*3387

*It really came from looking for 0.025 in the chart on the next slide.*3393

*What we are doing here is, I just want to justify to you, *3401

*we wanted the probability that Z is bigger than some little cutoff value of Z to be 0.025.*3404

*That is what we figure out on the previous slide.*3415

*I’m looking for 0.025 in the chart here.*3418

*It looks like these numbers are getting smaller and smaller.*3421

*I’m going to keep looking through these numbers.*3429

*Here, I’m getting close 0.28, .0281, .0274, .0265, 0.0262, .0256, .0250.*3430

*I found it, there is my answer right there.*3441

*I’m going to read off what row and column those came from.*3444

*It came from 1.9 and 0.06, that means that my Z value, my z is 1.96.*3448

*That is where that number came from.*3460

*I will say, we use that number, use on the previous slide,*3465

*and we did some work calculations with that to derive that we want N to be 11, was the answer that we got.*3474

*N = 11 samples.*3483

*You can go back and watch the previous slide, if you do not remember where that came from.*3486

*I would not go over that again now, you can just watch it again if you like.*3492

*What we did on this slide was, we are looking for that cutoff that gave us a tail probability.*3495

*Remember, this tail probability is what we are looking for, that was supposed to be 0.025.*3500

*The real reason for that was, that would make the other tail probability 0.025.*3508

*When you take those 2 probabilities away from 1 ,you get in the middle the probability is 0.95 which is what we are looking for.*3514

*That is where we got the 0.025 from.*3523

*But, we need to figure out which cutoff gave us that probability.*3526

*I found 0.025 in the table, read off its numbers 1.9 and 0.06.*3529

*I got Z = 1.96, and then I did some more calculations with that on the previous slide, to get down to N =11 samples.*3535

*That wraps up this lecture on sampling from a normal distribution.*3545

*The next lecture is going to look very similar to this, but we are going to be using the central limit theorem.*3550

*All the examples will have very similar flavor, we are sort of converting to*3555

*a standard normal variable then looking things up in the charts.*3560

*But, the difference is we are going to be using the central limit theorem *3564

*which means we would not have to start with a normal population anymore.*3567

*When we use the central limit theorem, you can start with any population in the world, *3571

*and then answer the same kinds of questions about whether your sample mean*3575

*is going to be close to your population mean.*3580

*I hope you will stick around and learn the center limit theorem.*3583

*It is probably one of the most important results in probability, that is in the next lecture.*3586

*That is also going to be our last lecture in the series, we are getting near the end.*3591

*I really appreciate you are sticking around me to enjoy these probability lectures.*3594

*This is the probability lecture series here on www.educator.com.*3599

*I am your host, my name is Will Murray, thank you for joining me today, bye.*3604

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