For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

### The Central Limit Theorem

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
- Our Samples are Independent (Independent Identically Distributed)
- No Longer Assume that the Population is Normally Distributed
- The Central Limit Theorem
- Standard Normal Distribution
- Converting to Standard Normal
- Example I: Probability of Finishing Your Homework
- Example I: Solution
- Example I: Summary
- Example I: Confirming with the Standard Normal Distribution Chart
- Example II: Probability of Selling Muffins
- Example II: Solution
- Example II: Summary
- Example II: Confirming with the Standard Normal Distribution Chart
- Example III: Probability that a Soda Dispenser Gives the Correct Amount of Soda
- Example III: Solution
- Example III: Summary
- Example III: Confirming with the Standard Normal Distribution Chart
- Example IV: How Many Samples Should She Take?
- Example IV: Solution
- Example IV: Summary
- Example IV: Confirming with the Standard Normal Distribution Chart
- Example V: Restaurant Revenue

- Intro 0:00
- Setting 0:52
- Setting
- Assumptions and Notation 2:53
- Our Samples are Independent (Independent Identically Distributed)
- No Longer Assume that the Population is Normally Distributed
- The Central Limit Theorem 4:36
- The Central Limit Theorem Overview
- The Central Limit Theorem in Practice
- Standard Normal Distribution 8:09
- Standard Normal Distribution
- Converting to Standard Normal 10:13
- Recall: If Y is Normal, Then …
- Corollary to Theorem
- Example I: Probability of Finishing Your Homework 12:56
- Example I: Solution
- Example I: Summary
- Example I: Confirming with the Standard Normal Distribution Chart
- Example II: Probability of Selling Muffins 21:26
- Example II: Solution
- Example II: Summary
- Example II: Confirming with the Standard Normal Distribution Chart
- Example III: Probability that a Soda Dispenser Gives the Correct Amount of Soda 32:41
- Example III: Solution
- Example III: Summary
- Example III: Confirming with the Standard Normal Distribution Chart
- Example IV: How Many Samples Should She Take? 42:06
- Example IV: Solution
- Example IV: Summary
- Example IV: Confirming with the Standard Normal Distribution Chart
- Example V: Restaurant Revenue 54:41
- Example V: Solution
- Example V: Summary
- Example V: Confirming with the Standard Normal Distribution Chart

### Introduction to Probability Online Course

### Transcription: The Central Limit Theorem

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

*This is our very last probability lecture, I’m want to say a special thank you*0006

*to those of you who stuck with me through all the videos.*0010

*Today, we are going to talk about the central limit theorem which is one of the crown jewels of probability.*0014

*I'm very excited to talk about the central limit theorem and show you how it plays a role in sampling.*0019

*We will be doing a lot of problems, solving questions about samplings.*0026

*I need to give you the background here.*0031

*It starts out just like the previous video.*0033

*If you watched the previous video, sampling from a normal distribution, then the first slide is going to be exactly the same.*0036

*You can safely skip that and then, I will show you what the difference is when we are using the central limit theorem .*0044

*Let us jump into that, the setting here, like I said, this is exactly the same as in the previous video, at least for the first slide.*0051

*The idea is that we have a population of stuff.*0061

*For example, we could have a whole bunch of students at a university and each student is a different height.*0063

*We have some distribution of heights at the university.*0071

*There are some population mean which means the average of all the students at the university.*0074

*We might or might not know that, that μ might be known or it might not be known.*0080

*There is some variance σ².*0085

*In the problems that we are going to solve today, we will need to know what the variance is.*0088

*That should be given to you in the problems.*0092

*And then, we are going to take some samples which means we are going to go out in the quad of the university.*0094

*We will stop some students randomly and survey them on how tall they are.*0102

*Or if we do not like measuring how tall people are, you can ask them how many units they are carrying *0107

*or how much student that they have, or what their bank balance is, or other GPA.*0113

*It does not really matter for the purposes of probability, what quantity we are keeping track of, *0119

*the important thing is that we are taking random samples.*0124

*The way we are going to keep track of them is, each student that we talked to counts as 1 random variable.*0130

*For example, if we are talking about the heights of the students then Y1 is the height of the first student,*0137

*Y2 is the height of the second student, and so on, until nth student.*0142

*If we want to survey N students then YN is the height of the last student.*0148

*Each one of those counts as a random variable and we will calculate the average of those samples.*0154

*We will talk about probability questions related to whether the average of our sample *0161

*is really close to the average of the entire population.*0166

*That is all the same as in the previous lecture.*0171

*What else is the same as the previous lecture is that our samples are independent.*0174

*There is this catch phrase that you hear in probability a lot and in statistics,*0179

*independent identically distributed random variables.*0184

*They are independent meaning that, if we meet 1 student and that student is very tall, *0188

*it does not really tell us that the next student is going to be tall or short because they are independent.*0194

*Identically distributed means they are all coming from the same population.*0200

*That is a buzz phrase to say independent identically distributed random variables.*0206

*Here is where this lecture is different from the previous lecture.*0212

*In the previous lecture, we have to assume that our population was normally distributed, *0216

*which is not really valid when you are talking about heights of students.*0221

*Because one thing, a student can never have a negative height so it is not really normally distributed.*0225

*In this lecture, using the central limit theorem, which I have not gotten to yet, *0232

*we do not have to assume that the population is normally distributed.*0236

*The beautiful thing about the central limit theorem is that the population could have any distribution at all.*0241

*The central limit theorem is very broad and it applies to any distribution at all.*0247

*In particular, when you are doing sampling, you do not have to know the general parameters of the population at all.*0254

*You do not need to know that your population is a normal distribution.*0261

*That is the key feature of the central limit theorem is that, *0265

*you do not need to know ahead of time what kind of distribution you are working with.*0269

*Let me actually tell you, what the central limit theorem is.*0275

*It says that, if you have independent samples from any population with mean μ and variance σ²,*0279

*these are IID independent identically distributed samples.*0287

*The conclusion of the central limit theorem says that, Y ̅ is the sample mean.*0292

*That means you take the samples that you collect and you take their average, *0300

*just of those samples, and that is a new random variable.*0309

*That is a function of the random variables you had before.*0313

*It says that, the distribution of that random variable approaches as N goes to infinity, *0316

*it gets closer and closer to a normal distribution with mean μ and variance σ²/N.*0324

*This is really one of the most extraordinary facts in all of mathematics, *0335

*which is that we did not assume that the original population was normally distributed.*0340

*But, even without that assumption, the sample mean approaches a normal distribution.*0347

*This is sort of why the bell curve, the normal distribution is considered the most important distribution in all probability and statistics.*0356

*It is because even if you do not start with the normal distribution, you could start with any distribution at all,*0365

*you always end up with a normal distribution, as you take samples and you look at the sample mean.*0371

*That is really quite extraordinary but the math is very powerful and it does work out that way.*0379

*Let me mention that in practice, it is kind of the rule of thumb that people use in practice*0386

*when applying the central limit theorem is that, it starts to kick in, remembering it applies as N goes to infinity.*0398

*It really starts to become useful when N is bigger than about 30.*0408

*N is the number of samples that you take.*0416

*If you take more than 30 samples, you can safely assume that your sample mean will follow a normal distribution.*0418

*We can invoke the central limit theorem and say that the sample mean will have a normal distribution, *0430

*a normal distribution with mean μ and variance σ²/N.*0446

*That is kind of how the central limit theorem is used in practice.*0451

*As long as you take at least of 30 samples then, you can say that your sample mean is going to have a normal distribution.*0456

*It does not even matter, what the distribution of your original population was.*0466

*What do we actually do with that, once we know that the sample mean has a normal distribution, *0471

*what are we supposed to do with that.*0476

*From then on, it is pretty much the same as in the previous lecture.*0478

*We can walk you through that, in case you did not just watch the previous video.*0483

*What you do with a normal distribution is you convert it to a standard normal distribution.*0488

*A standard normal distribution, I will remind you is a normal distribution with mean 0 and variance 1.*0494

*It is what it means to be a standard normal distribution as mean 0 and variance 1.*0507

*The point of the standard normal distribution is that, you can look up probabilities for a standard normal distribution using charts.*0513

*Of course, there also lots of online applets that you can use, *0521

*a lot of computer programs will know tell you probabilities for a standard normal distribution.*0525

*For the videos in this lecture, I'm going to use charts.*0531

*If you are lucky enough to have access to the some kind of online tool or computer program, *0536

*that will tell you probabilities for a standard normal distribution then by all means, have at it and use that.*0541

*This is kind of an archaic method that I'm showing you here.*0550

*But still, in a lot of classroom settings people still use chart that is why I’m showing it to you.*0553

*That is how you can look up probabilities for a standard normal distribution.*0558

*This picture is one that is really useful to keep in mind.*0563

*This chart, the way it works is, it tells you the probability of being above a certain cutoff.*0570

*If you want to find the probability of being less than that cutoff, then you have to do something like subtracting from 1.*0577

*If you want to be between 2 cutoffs, then you have to figure out the probability of being in the tails*0583

*and then subtract those from 1.*0589

*That is a kind of computations that you have to do to use these charts.*0591

*That is all based on a standard normal distribution.*0596

*In practice, you usually do not get a standard normal distribution.*0602

*You usually get some kind of random normal distribution.*0606

*Let me show you how you convert it to a standard normal distribution.*0609

*I say recall because I did a whole lecture on this, earlier on in these probability lectures.*0615

*If this is totally new to you, what you might want to do is go back *0621

*and work through the lecture on the normal distribution, that we studied earlier on in this lecture series.*0624

*But if you already worked through that lecture, maybe you just need a quick refresher, here you go.*0631

*What we learned back then was that, if Y is any normal distribution then *0635

*what you can do is convert it to a standard normal distribution,*0642

*by subtracting off the mean and dividing by its standard deviation.*0645

*We call that new variable Z, and what we learned is that Z is a standard normal distribution.*0651

*The point of getting a standard normal distribution is then,*0660

*you can look up probabilities in terms of Z and convert them back to find probabilities in terms of Y.*0662

*What we learned in our central limit theorem was that, Y ̅ is essentially, *0670

*it approaches a normal distribution with mean μ and variance σ²/N.*0679

*What that means is that, if we do Y ̅ and we subtract its mean, that is Y ̅ - μ and divide by,*0687

*its standard deviation is always the square root of its variance.*0697

*I have to do √ σ ⁺N/N, I’m going to call that Z.*0702

*Now, you notice if I take the denominator and flip it upside down because of the fraction in the denominator,*0710

*then I will get σ², √σ² is just σ.*0719

*√N is going to flip up to the numerator, that is why I get that √N/σ × Y – μ.*0726

*That is where I’m getting this expression right here, that is where that comes from.*0734

*The variable that we just created is a standard normal variable.*0741

*I can use the charts to look up probabilities for that standard normal variable.*0745

*That is how I’m going to be solving the examples.*0751

*I’m going to ask you some kind of question about Y ̅, *0755

*and then what we will do is we will build up the standard normal variable, translate it into a question about Z.*0760

*And then, we will use the charts to look up probabilities on Z, that is how that is going to work.*0768

*Let us jump into the exercises and practice that.*0775

*In example 1, this is a very realistic problem.*0778

*Homework problems take you an average of 12 minutes each, but there is a lot of variation there,*0782

*there is a standard deviation of 10 minutes.*0787

*Maybe, if you get a real quick problem, you can quickly dispense of it in 2 minutes.*0790

*Or if you get a really tough one, it could take you 22 minutes or possibly even longer.*0797

*Your assignment is to solve 36 problems, this is going to take awhile.*0802

*What is the probability that it will take you more than 9 hours?*0806

*Let us think about that, first of all I have a conversion to solve here.*0811

*9 hours is 9 hours × 60 minutes per hour, that is 540 minutes.*0819

*If I'm going to take 540 minutes, I want to convert that into an average .*0838

*Remember, what I want to use is that Z is √N/σ × Y ̅ – μ.*0845

*I know that, that will be a standard normal variable.*0858

*Somehow, I got to buildup those quantities.*0861

*If my total time spent on the homework is going to be 9 hours, that is 540 minutes.*0864

*How much time will that be per problem on average?*0871

*My Y ̅, which is the average time per problem, if I spent 540 minutes total then *0875

*that would be 540 minutes divided by 36 problems.*0888

*I rigged up those numbers to work fairly nicely.*0894

*That is 15 minutes per problem.*0898

*I want to know the likelihood that I'm going to end up spending more than 15 minutes per problem,*0909

*over a 36 problem assignment.*0915

*I want to find the probability that my Y ̅ is bigger than 15.*0922

*Somehow, I want to build up this standard normal variable so I can use my normal distribution charts to solve this.*0935

*If Y ̅ is bigger than 15, that mean Y ̅ - μ is bigger than, what was my μ?*0942

*My μ is the average of all homework problems, 12 minutes on average,*0951

*is what it takes me to solve a homework problem.*0958

*15 -12, I will put in a σ and what is my σ?*0962

*My σ is standard deviation, that is 10.*0973

*The last ingredient here is √N, √N I'm going to include that.*0980

*What is my √N, N is the number of problems that I have.*0991

*N was 36, the √N is 6, this is 3 × 6/10, 18/10 is 1.8.*0995

*That was my variable Z, I want to find the probability that Z, my standard normal variable is going to be bigger than 1.8.*1007

*I'm going to look that up on the next page because I have a standard normal chart all set to go, on the next page.*1021

*Let me just go ahead and tell you the answer, so we can wrap it up on this page.*1030

*From the chart on the next page, and I will show you in a moment where that comes from.*1035

*The probability what was it, it was 0.0359 is what we are going to find on the next page.*1040

*If I want to think about in terms of percentages, that is just about 3.6%.*1052

*That is my probability that I'm going to spend more than 9 hours on this homework assignment.*1060

*It was about 3.6% chance that I'm going to spend more than 9 hours on this homework assignment.*1068

*Maybe, I’m worried because I have something I need to do in 9 hours.*1074

*I’m worried I would not get finished in time.*1076

*It actually looks pretty good, it looks like there is more than 96% chance that I will finish on time, that is kind of reassuring.*1080

*That is the answer to the problem, there is 3.6% chance that we will spend more than 9 hours on this homework assignment.*1088

*Let me just recap the steps there, before I show you that one missing step of looking it up on the chart.*1095

*We want to, first of all, convert into a standard unit here.*1102

*I got 9 hours and I got 12 minutes, I decided to convert the hours into minutes.*1107

*You can also convert the other way, if you wanted, but I think it is a little easier this way.*1114

*9 hours is 540 minutes, that was a total on our time that I would spend doing all the homework problems.*1119

*Since, I know that I have a result about averages here, I wanted to convert that into an average amount of time.*1128

*The average time is 540 minutes divided by 36 problems and that is just 15 minutes per problem.*1138

*The question is really, how likely am I to spend an average of more than 15 minutes per problem?*1148

*I want to find the probability that Y ̅ is bigger than 15.*1154

*For Y ̅ to be bigger than 15, I want to build up this expression Y ̅ - μ/σ × √N.*1159

*I filled in my μ is 12, μ right there is the average.*1168

*My σ was 10, there is σ and I got that from the problem here.*1176

*My √N is 6, that comes from N = 36, the number of problems that we have to solve here.*1185

*Then, I just simplified the numbers 15 -12 is 3, 3 × 6 is 18, 18 divided by 10 is 1.8.*1193

*I want to find the probability that my standard normal variable is bigger than 1.8.*1199

*That is what I’m going to confirm on the next page.*1205

*I will show you where that comes from, but we will see that it comes out to 0.0359.*1207

*If you think about that as a percentage, that is just about 3.6%.*1214

*I just want to confirm the result that we used on the previous page.*1225

*What we use on the previous page was, we calculated the probability that a standard normal variable, *1231

*because we had converted a Y ̅ to a standard normal variable, was bigger than 1.8.*1236

*I’m going to look up 1.8 on this chart, 1.80.*1245

*There is 1.8, there is 1.80, it is 0.0359 which was the number that I gave you back on the previous slide.*1249

*This shows you where that come from, it just come from this chart.*1260

*.0359, that is where I got that answer of 3.6% that we used on the previous slide.*1263

*That just completes that little gap that we had on the previous side.*1274

*That totally answers our probability of having to spend more than 9 hours on this horrible homework assignment.*1278

*In example 2, we have a bakery that is charting how many muffins do they start per day.*1288

*We have figure out a long-term average of 30 muffins per day but there is a lot of variation in there.*1295

*Maybe, they sell more muffins on the weekend and fewer on a weekday.*1300

*They figure out that, there is a standard deviation of 8 muffins.*1305

*What they are doing is, they are planning out the next month or so, actually 36 days.*1309

*They are worried about, what is the chance that they will sell more than 1000 muffins?*1315

*Maybe, they are worried about whether they are going to have to order some more supplies,*1320

*some more flours, some more eggs, or something like that.*1323

*Or maybe, they are worry about whether they are going to make enough money.*1327

*They know they need to sell 1000 muffins in the next 36 days.*1330

*It was the kind of calculations that a business person would make.*1334

*We are going to answer them using the central limit theorem.*1338

*Let me remind you what we have, our mean theorem is that, if we start out with Y ̅ – μ, very important distinction there.*1342

*Y ̅ - μ × √N/σ is a standard normal variable, that is kind of our main result for this lecture.*1357

*We want to figure out how we can use that here.*1371

*I see a Y ̅ there, that Y ̅ is the average of the number of muffins we are going to sell each day.*1375

*If the total muffins is going to be 1000, then that means the daily average is Y ̅ which will be 1000/36,*1384

*because there are 36 days.*1412

*It does simply a bit, I can take a 4 out of top and bottom there, simplify that down to 250/9, *1413

*still not the nicest fraction in the world.*1421

*I'm going to try to build up the standard normal variable and get an answer that I can look up easily on the normal charts.*1425

*Y ̅ - μ is 250/9 - μ is the overall average which we figure out is 30 muffins per day, that is -30.*1434

*That is a little awkward, let me go ahead and try to combine those fractions.*1451

*I did do this one in fractions because I rigged it up so the fraction work fairly nicely, *1457

*something we can work out in our heads.*1462

*If the fractions did not work nicely, I would probably just be going to a decimal right now.*1464

*But, this one works nicely.*1468

*250/9 - 30 is 270/9, we get -20/9, you can convert that into a decimal, if you like.*1471

*Let me continue to build up the standard normal variable.*1486

*√N × Y ̅ - μ/σ, I want to be bigger than the values that we have, because we want to sell more than 1000 muffins.*1490

* This should have been a greater than or equal to.*1507

*This should be greater than or equal to -20/9.*1512

*I’m multiplying by √N, the √N is 36, that is because N is 36.*1517

*√36 is 6, and now I have a σ.*1522

*Σ is our standard deviation, that is 8.*1531

*That is because, what we are told in the problem here.*1537

*I think this does simple fairly well, this is - 20/8 could simplify to 5/2, 6/2 could simplify to 3/1.*1542

*3/9 could simplify to 1/3, we just get -5/3.*1557

*Now, I'm going to convert it into a decimal.*1563

*The point of this was that, this was a standard normal variable.*1566

*I want that Z to be bigger than or equal to -5/3 which is as a decimal is -1.67.*1573

*2/3 is about 0.67, most technically that is an approximation.*1580

*I do not want any pure mathematicians to complain about that.*1587

*It is -1.67, and now I want to figure out the probability that Z will be bigger than -1.67.*1591

*Let me draw what I'm going to be looking for, then we will use a chart on the next page to actually calculate that.*1602

*-1.67 was down here somewhere.*1610

*I’m looking for the probability that Z is bigger than that.*1617

*I’m looking for all that probability.*1619

*The way the chart works is it will tell me probabilities of being bigger than a certain cutoff.*1622

*What I'm going to do is find the probability that Z is bigger than 1.67 and then subtract that.*1630

*The probability that Z is bigger than -1.67 is going to be 1 - the probability that Z is bigger than +1.67.*1639

*This is what we are looking for, this probability that Z is bigger than -1.67.*1660

*But I can figure it out as 1- this probability, the probability that Z is bigger than 1.67.*1670

*That is how I'm going to calculate that out.*1681

*The rest of it is simply a matter of looking it up on the chart, because that is the form that I can look things up on the chart.*1684

*I have already done this, if I look it up on the chart on the next page.*1690

*Let me tell you what the answer comes out to be.*1696

*It is 1- 0.0475 and that comes from the chart on the next page.*1699

*And then, 1- 0.0475 is 0.9525, and that is approximately 95%.*1708

*It is in fact very likely that, this bakery is going to sell more than 1000 muffins in the next 36 days.*1721

*If they are planning on buying supplies, buying flour, eggs for their muffins, *1730

*then they better go out and buy more supplies because it is very likely that they will sell more than 1000 muffins.*1735

*If they are worried about revenue then things are looking pretty good,*1742

*because there was a good chance that they will sell more than 1000 muffins.*1745

*Let me recap the steps here.*1750

*There is one missing step which is the chart, which I will fill in on the next slide.*1751

*In the meantime, the total muffins, we want that to be bigger than 1000,*1757

*which means the daily average should be bigger than 1000/36, which is more than 250/9.*1762

*I was just reducing the fractions there.*1770

*I rigged this one up to give us nice fractions.*1772

*And then, I kind of built up this standard normal variable.*1775

*I subtract a μ, the μ was the average of 30 muffins per day, that comes from there.*1779

*That is where that 30, and subtracted it and I got a negative number.*1787

*It is significant that it is negative there.*1792

*We do want to keep track of the negative sign.*1795

*And then, I multiply by √N which was, there is my N is 36.*1798

*√N is my 6 right there, divided by σ which is the standard deviation, 8 muffins right there, there is my 8.*1804

*And then, I just did some simplifying fractions there, got down to -5/3.*1813

*And I convert that back into a decimal which is -1.67.*1820

*In order to find the probability of Z being bigger than -1.67, I flipped it around and I calculated that the probability of being in this tail.*1827

*That is the probability of being in the tail, the probability that Z is bigger than 1.67.*1840

*For that, I’m going to use the chart on the next page.*1847

*I hope I have been reading the chart correctly.*1849

*When we look on the next page, it really will be 0.0475.*1852

*It will work out then to be 1- that is 95% chance that this bakery will sell more than 1000 muffins.*1856

*That is just filling that one missing step from the chart.*1867

*This is the normal distribution chart, this will tell you the probability that in normal variable, *1871

*we will end up being bigger than a particular cutoff.*1878

*In this case, our cutoff is 1.67, we are using that to solve the problem on the previous slide.*1881

*The probability that Z is bigger than 1.67, here is 1.6 and the second decimal place is 7, it is right there.*1887

*I hope this works out, 1.6 and 0.0475, that is what we use before.*1899

*It is 0.0475 and that was the answer that we plugged into our calculations on the previous side.*1907

*Just take this answer, drop into the calculations on the previous side.*1918

*And then, we did some more work and we got our answer to be 95%.*1923

*That fills in the one missing step from the previous slide.*1932

*It is just a matter of taking this 1.67 and matching up 1.6 and 0.07, and finding the right probability.*1938

*Of course, if you are using electronic tools, you probably do not need to use this chart.*1948

*You can just ask what is the probability that a standard normal variable will be bigger than 1.67, *1953

*and it should just spit out the answer for you.*1959

*In example 3, we have a technician fixing a soda machine.*1963

*It is supposed to give a certain amount of soda and she wants to check out whether it is dispensing the right amount of soda.*1969

*She takes 100 samples and the standard deviation in this machine is 2.5 ml.*1976

*We want to find the chance that her sample mean, that is the Y ̅, *1986

*that is the mean of her 100 samples is within 0.5 ml of the true average amount.*1990

*That is the true population mean, that is the μ of soda dispensed.*1997

*Let me setup that one out for you.*2002

*The whole point of this lecture is that, we look at Y ̅ - μ and we convert that to a standard normal variable.*2005

*The way we do that is by multiplying by √N/σ.*2016

*That turned out to be a standard normal variable meaning it has mean 0 and standard of deviation 1.*2022

*What we want to is figure out Y ̅ – μ.*2032

*In this case, we want the sample mean to be within 5 ml of the true mean.*2038

*Within 5 ml means it could go 0.5 ml either way.*2044

*I’m going to put absolute values here and set it less than or equal to 0.5 ml.*2051

*I’m going to try to build up my standard normal variable.*2061

*I'm going to build up by putting √N here and dividing by σ.*2065

*Of course, I got to do that on the other side, as well, √N and divided by σ.*2071

*The point of that is, that gives me a standard normal variable or actually there is a mass of values their, *2078

*I put the absolute values on Z as well.*2084

*That means, I want the absolute value of Z less than or equal to, let me fill in what I can here.*2087

*0.5 √N, N is 100 because we are taking 100 samples here.*2093

*That is × 10, √100 is 10, I rigged that up to make the numbers easy.*2101

*What is my σ, σ is 2.5 here.*2108

*10 divided by 2.5 is 4, this is 0.5 × 4 which is 2.*2117

*That worked out really nicely, is not it.*2125

*I want the probability that the absolute value of Z will be less than 2.*2127

*It is the same as the probability that Z is between -2 and 2.*2134

*Let me draw a little picture, it is always useful when you are working with these normal distributions*2146

*to draw a picture and figure out what it is you are actually calculating.*2150

*I want Z to be between -2 and 2, there is -2 and 2.*2155

*I want to be in between there.*2162

*The way my chart is set up, your chart might be different, but the way my chart works is, *2164

*it will tell you the probability of Z being in the positive tail.*2170

*It will tell you that area right there.*2175

*In order to find the probability in the middle, what I'm going to do is *2179

*calculate that probability in the tail and then subtract off 2 tails, that will give me the probability in the middle.*2184

*I will do 1-2 × the probability that Z is bigger than 2.*2192

*That should give me the probability of being between -2 and 2.*2199

*That is something now that I can look up on my chart,*2205

*or if you do not like charts and you got access to some kind of electronic tool, you can look that up more quickly.*2209

*Just drop the number 2 into your tool.*2216

*I’m going to use the chart and it is on the next page where I set up the charts.*2219

*Let me go ahead and tell you the numerical answer now, and then let me just confirm that on the next page.*2224

*I found the probability of Z being bigger than 2 from the chart.*2229

*It was 0.0228, and that is 1- 2 × 0.0228 is 0.0456.*2234

*That is 0.9544, and that is approximately 95%, just slightly over 95%.*2250

*That is my answer, that is the probability that this technician’s sample mean will be within 0.5 ml of the true mean.*2271

*Let me recap the steps there.*2286

*There is one missing step which is the step from the chart, which will confirm that on the next slide.*2288

*Just while we have the slide in front of us, let me recap the steps.*2295

*I set up my standard normal variable, I know that Y ̅ - μ is always × √N/σ is a standard normal.*2299

*I wanted Y ̅ - μ to be within 0.5 of each other.*2311

*Y ̅ μ should be within 0.5 of each other.*2317

*When we say two things are within a certain distance from each other, that really means that you want to bound their absolute value.*2320

*The distance from A to B is the absolute value of A – B.*2329

*We want the absolute value of Y ̅ - μ to be less than 0.5.*2333

*I just tacked on these other quantities, √N/σ.*2338

*The point of that was that gave me a standard normal variable, I can call that Z.*2343

*And then, I wanted to fill in what my √N and σ were.*2350

*N was 100, √N gave me 10, that is where that came from.*2354

*The σ was the standard deviation given there, that is my σ 2.5.*2360

*The numbers simplified nicely, that was me being clever, setting up nice numbers there.*2367

*It simplified down to 2, but now you have to think a bit more because you want the probability that Z is less than 2.*2374

*Absolute value, that means Z is between -2 and 2, that is the middle region here.*2383

*The way my chart works, it will tell you the tail region, that would not you the middle region directly.*2391

*What I did was, I said you could solve this by doing 1-2 tails, because there is a lower tail and there is an upper tail there.*2397

*I'm going to look up on the chart on the next page and we will see that the probability of Z being bigger than to is 0.0228.*2407

*It is 1-2 × that, 2 × that is 0.0456.*2416

*We finally get 0.9544, get a 95% chance that we will be within 0.5 ml.*2421

*By the way, example 4 is a follow up to example 3.*2430

*I want to make sure that you understand example 3.*2435

*It there is any steps in here that you are a little fuzzy on, just watch the video again and *2437

*just make sure that you are very clear on all the steps here,*2442

*because in example 4, we are going to tweak the numbers a little bit.*2445

*It really will help if example 3 is already very solid for you.*2449

*One missing step here is where that number comes from.*2453

*Let us fill in that step, this is still example 3 and we had one missing step *2457

*which was the probability that Z was bigger than 2, that was from the previous side.*2464

*In order to solve that, I see that there is 2.0 right there.*2471

*2.00 gives me 0. 0228 is what I used on the previous side.*2477

*I use that number on the previous side and you can catch up with all the rest of the arithmetic on the previous side.*2492

*If you do not remember how that worked out but we did some calculations with that.*2501

*We came up with an answer of 95% for this soda dispensing machine.*2505

*That wraps up example 3, we are going to use the same scenario for example 4.*2514

*I do want to make sure that you understand example 3, before you go ahead and try example 4.*2519

*In example 4, we have the same technician from example 3.*2528

*Remember, this technician is taking samples from a soda dispensing machine.*2532

*She wants to guarantee with probability 95% that her sample mean will be with 0.4ml of the true average.*2537

*This looks a lot like example 3, the difference, I will just remind you was with examples 3, *2547

*we had not 0.4 but a 0.5 ml tolerance.*2555

*Here we are restricting that to 0.4 ml tolerance.*2562

*We are deciding that 0.5 is not close enough, I want to get a 0.4 ml tolerance.*2566

*I still want to keep the probability at 95%.*2572

*That means, I have to change something else.*2576

*Since, I want a more accurate answer, that means I'm going to need to take more samples than I did before.*2580

*We are going to try and solve that together.*2587

*First thing is to figure out with this probability 95%, what is that mean?*2591

*Let me try to illustrate that graphically.*2597

*I want to find some cutoff that gives me 95% of the probability in the middle.*2602

*I will put a value of Z there, that is –Z.*2611

*I want to get 95% of the probability in between these cutoffs, for the standard normal variable.*2614

*This is supposed to be 95% here and I want to figure out what value of Z will give me that.*2621

*The way to find that, since I have a chart that will tell me how much probability is in the tail of the normal distribution, *2631

*what I can do is say that these tails, I got two tails here, *2644

*the area in 1 tail should be 1- 95%, 1 - 0.95/2, which is 0.05/2 which is 0.025.*2649

*I want to find a Z value, since that the probability of Z being bigger than that cutoff z is 0.025.*2668

*I will check this on the next slide, you will see, we will look it up together.*2680

*I do not want to break my flow for this slide, I will tell you right now that it comes out to be Z = 1.96.*2684

*I want to make sure that was the right value that I used.*2694

*Yes 1.96, that is the cutoff Z value that we are looking for.*2696

*Let me go back and show you how that factors in with all the other numbers in the problem.*2702

*I will just remind you that Z is our standard normal variable.*2706

*The way we get it is we do √N/σ × Y ̅ – μ.*2711

*Let us work that, I wanted out to have Y ̅ – μ.*2720

*I wanted those two quantities, Y ̅ is the sample mean, that is Y ̅ right there.*2726

*The true average of this machine is μ, I do not know what that is, by the way.*2733

*I want this to be within 0.4 of each other.*2739

*I want the difference between those two in absolute value to be less than 0.4.*2741

*When I want to do is build up this standard normal variable.*2755

*I’m going to multiply on √N/σ here.*2760

*I will multiply on √N/σ, as well here.*2764

*The point of that is this gives me my Z value, my standard normal variable.*2771

*My Z is less than or equal to √N × 0.4.*2779

*I know what σ is, my σ was given to me in example 3, that was 2.5.*2786

*That is 2.5 from example 3 was where that was given to us.*2793

*Let me fill that in, example 3 was where that information came from.*2799

*The N in example 3, N was 100 because we took 100 samples.*2805

*We cannot use that anymore because now we are trying to go for more accurate estimation.*2810

*We are going to have to increase the number of samples.*2817

*I know it is going to be increased because we have to get a more accurate answer.*2820

*What we are really going to do is solve this for N.*2823

*We are going to use this value of Z that we figured out over here, 1.96, and then we will solve for N.*2831

*1.96 is less than or equal to √N × 0.4/2.5.*2841

*Now, I’m just going to manipulate the algebra a little bit until I can get a value for N.*2852

*I do not think this one is going to work out particularly nicely, I did not rigged the numbers for this one very well.*2860

*2.5, I will multiply by both sides, I will divide both sides by 0.4, that should still be less than √N.*2865

*N should be greater than or equal to 1.96 × 2.5 divided by 0.4.*2876

*Since, I changed from √N to N, I'm squaring both sides.*2888

*We should square that expression.*2893

*That is not a number you will have work out in your head.*2897

*I did that on a calculator and I will show you what I got there.*2901

*I got 150.063, just slightly over 150 there.*2905

*N is the number of samples that we are going to take.*2916

*It is got to be a whole number, because you cannot take half of the sample.*2919

*In order to make this work, I need a whole number bigger than 150.6063.*2923

*N = 151, I will round that up.*2931

*You always round up, if you are talking about the number of samples.*2936

*The samples is enough, I solved that except for one detail of showing you on the chart where that Z value came from.*2940

*I will go back over the steps here and then we will jump forward to the chart.*2952

*I will show you where that Z value came from.*2957

*To go back to be getting here.*2960

*I start out with this probability of 95%.*2964

*I’m going for a 95% probability here.*2967

*In order to get 95% in the middle, that means my values on the tail, there is two tails, they are going to split the left over.*2972

*The left over probability is 1 - 0.95/2.*2982

*1-0.95 is 0.05, .05/2 is 0.025.*2988

*I’m looking for cutoff value of z, that when I find the probability bigger than that, it is 0.025.*2994

*It will come from the chart on the next slide.*3003

*L will fill the part in, if you are willing to be a little patient with me, or you can skip ahead and see the chart on the next slide.*3007

*Then, I would just going to hang onto that Z for a little while.*3015

*I'm going to go back and I'm going to build up my standard normal variable here, that comes from this formula here.*3018

*That is supposed to be a standard normal distribution.*3023

*I started off with Y - μ and their absolute value, that comes from the word within here.*3029

*I want two things to be within 0.4 ml.*3035

*I want the absolute value of less than 0.4.*3039

*And then, I built up my √N/σ.*3042

*I do not know what the √N is because I have not been told how many samples I want at this point,*3046

*that is what I'm solving for.*3051

*The N is the variable but my σ, I was given the standard deviation in example 3.*3054

*I dropped that in, it is 2.5.*3059

*I fill in my Z value, that comes from over here.*3062

*That is where that Z value came from.*3066

*I just take this equation and I solve it for N.*3071

*That is just a matter of manipulating the algebra around, squaring both sides because I had a √N.*3075

*I get N = 150.063, and I need it to be a whole number.*3080

*To be safe, I round it up.*3089

*Because 150 samples would not be quite enough, I have to go for 151 samples.*3092

*And then, I know with probability 95% that my sample mean will be close to the true mean.*3098

*One missing step here is where that 1.96 came from and it comes from this 0.025.*3106

*But, I'm using the chart that we will see on the next slide.*3114

*I want to fill in that one missing step from example 4.*3118

*We want to find a Z for which the probability of Z being bigger than that Z is 0.025.*3122

*We work that out from the previous slide, that was what we are looking for.*3130

*Let me draw a little picture of what we are dealing with.*3139

*We want to find a Z, says that tail probability there is 0.025 which means I have to look for 0.025 in my chart.*3143

*I’m going to start here, they are getting smaller 0.7, 0.6, 0.5, 0.4, 0.3, 0.02.*3158

*It is getting close, 0.027, 0.026, 0.0256, 0.0250, there it is right there.*3172

*I look at where row and column that happened in, 1.96 and 0.06, that tells me that my Z value is 1.96.*3182

*You might also have electronic applets, you do not have to do this kind of old fashioned method of looking up on charts.*3197

*That is totally fine with me, if you are okay using electronic tools in your class.*3205

*You can jump from 0.025 to 1.96, that is totally fine with me.*3211

*This 1.96, we went on and use that in our calculations on the previous side.*3217

*Somehow, that work out on the previous side to tell us that we need N = 151 samples,*3228

*in order to guarantee a particular accuracy of our sample mean.*3237

*Just to recap there, most of the work was done on the previous side.*3245

*I figure out on the previous side that I was looking for cutoff value of Z, *3249

*such that the probability of being bigger than that was 0.025.*3253

*We just look through this chart until I found 0.025, found that in the 1.9 row and the 0.06 column.*3257

*I put those together and I get 1.96.*3265

*The rest of it goes back to the previous side where we threw that 1.96 in a bunch of calculations*3268

*and came back to N = 151 samples.*3275

*In our final example here, we have a restaurant that is worried about how much money it is going to make tonight.*3282

*It has done some studies and it is found that its customers spend an average of $30.00 per customer.*3294

*But, they have a standard deviation of $10.00 which means,*3301

*maybe if somebody just has an appetizer and a drink, maybe they will spend $20.00.*3303

*Maybe, if they really go for the full menu and have drinks and desserts, and a few different extras,*3308

*then they are going to end up spending $40.00 or even more.*3317

*The restaurant, their average is $30.00 and they have 25 reservations tonight.*3320

*I guess the reservation only restaurant, you cannot just walk in here, you have to have a reservation.*3326

*They are expecting 25 customers tonight.*3331

*They want to know the chance that their total revenue tonight, we are not worried about profit,*3334

*we are not worried about what we are spending on supplies, total revenue will be between $725 and $800.*3338

*Let me show you how this turns into a central limit theorem problem.*3348

*Because it is not totally obvious right now, we are talking about total revenue.*3353

*Let me show you here, let me remind you of what we are given at the beginning of this lecture.*3356

*We are given that Y ̅ is sample mean – μ the global mean, divided by σ the standard deviation, and multiplied by √N.*3361

*N is the size of the sample.*3378

*The point of that was that would give you a standard normal variable.*3381

*We are going to call it Z and that is a standard normal variable.*3386

*In turn, the point of a standard normal variable is it is very easy to calculate probabilities.*3392

*The way I’m doing it is I'm using charts.*3397

*You might use charts for your class or you might have more sophisticated electronic tools, and that is okay with me.*3400

*What is this have to do with this restaurant?*3407

*They want to make between $725 and $800 total.*3409

*If the total is going to be between 725, I said they want to make between that.*3416

*Of course, they would be happy if they made more.*3425

*They are worried about, they want calculate how likely is it that they will make between 725 and 800 total.*3428

*What we have here is a result that has to do with the mean, the sample mean.*3434

*How do we convert that into a mean?*3440

*We just divide by the customers, the number of customers,*3442

*and convert that into an average amount that each customer would spend.*3448

*The mean, the average, Y ̅ would have to be between 725 divided by 25 and 800/25*3453

*because that is how much the average customer would have to spend, in order to get the total between 725 and 800.*3468

*I just did a little arithmetic here, I rigged this up so that the numbers came out fairly nicely.*3476

*800/25 is 32 and 725/25 is 28, 725/25 is 29.*3482

*Y ̅ would have to be between 29 and 32. *3495

*What that means is, all the customers that come in tonight, *3501

*they would have to spend an average of between $29.00 and $32.00.*3504

*It does not mean they will have to spend between that, *3510

*but it means you can still have some big spenders to come in and drop 50 bucks on a meal.*3512

*You can still have some cheapskates to just buy an appetizer and then slid out of there after spending $10.00.*3517

*But on average, it has to come out between $29 and $32.00 per customer for tonight's customers.*3523

*What I would like to do is kind of buildup that standard normal variable.*3534

*Y ̅ - μ would be between, my μ is my global average.*3540

*That is right there, that is the 30, that is how much a customer spend on the average in the long term.*3547

*That is 29 -30 and 32 -30, let me go ahead and divide by σ.*3554

*My σ is the standard deviation, there it is $10.00 right there.*3569

*I also need to multiply by √N, I did not really leave myself enough space to do that.*3577

*I will give myself another line there.*3583

*Y ̅ - μ/σ × √N, this not absolute value, this is not one of those within problems like the previous one.*3584

*We have to be careful about what is positive, what is negative, no absolute value here.*3595

*√N, N is 25, that is the number of customers we are going to be working with tonight.*3599

*√N is 5, 5 × 29 -30/10 and 5 × 32 -30/10.*3604

*That simplifies fairly nicely, 5/10 is ½, ½ × -1 is - ½.*3619

*I will write that as -0.5.*3628

*And then, the point of this was we are building up a standard normal variable.*3631

*That is my Z right there, this is between -0.5 and 5/10 is still ½, 32 -30 is 2, 2 × ½ is just 1.0.*3635

*I have a standard normal variable and I want to find the probability that it is between -1/2 and +1.*3651

*Let me draw a graph of what I’m looking for.*3659

*Possibly, if you have the right electronic tool, you can jump to the answer at this point.*3663

*Just drop these numbers in your electronic tool, but let me show you how you can figure out using your charts.*3668

*There is -0.5 and there is 1.0.*3675

*I want to find the probability of being in between those two.*3680

*What my chart will do is, it will tell me the probability of being in a tail.*3685

*It will tell me that probability right there.*3690

*It will also tell me that probability right there, but those are not the same because 0.5 and 1.0 are not symmetric.*3693

*This is the probability that Z is greater than 1.0.*3701

*This is the probability that Z is less than -0.5, but it is also the same as Z being bigger than 0.5.*3707

*What I really want here is, my probability that Z is between -0.5, 0.5, and 1.0.*3716

*Let me write that a little more clearly, - 0.5 and 1.0*3730

*What I can do is, I can subtract off the two tails to get the probability.*3741

*That is 1- the probability that Z is bigger than 0.5 - the probability that Z is bigger than 1.0.*3746

*We have some other problems that is like this where we subtracted off the two tails.*3758

*Those are symmetric ones, they started out with absolute values.*3761

*We can just find one tail and then multiply it by 2.*3765

*But these tails are not symmetric, I would have to do two separate calculations and*3769

*look up two separate numbers on a chart there.*3775

*Let me say that I will do the steps on the chart on the next page.*3780

*I will just tell you what the answers are for now, and then I will prove to you by showing you the chart on the next page.*3785

*The probability that Z is bigger than 0.5, I look that up on my chart, I got 0.3085.*3793

*The probability that Z is bigger than 1.0 was 0.1587.*3804

*Now, it is just 1 - 0.3085 0.1587.*3812

*I was lazy, I threw that into a calculator and what I got was a 0.5328, *3817

*could have done that by hand, that would have been that bad.*3827

*If you want to estimate that, that would be just 53%.*3832

*That is the probability that this restaurant, their total profit for tonight is between $725 and $800 tonight.*3840

*There is that one missing step from the chart.*3853

*We will confirm that on the next slide but before I turn the page from the slide, let me show you the steps here.*3856

*I want to find the probability that my total was between 725 and 800.*3862

*What I really know is a result about the average that each customer tonight is going to spend.*3867

*I want to convert that total into an average.*3875

*I just divided it by the number of customers.*3878

*The total divided by the number of customers gives me the average.*3883

*725/25 gives 29, 800/25 gives me 32.6*3887

*And then, I start to build up this formula for a standard normal variable.*3894

*I subtracted μ from both sides, that μ was the average that all the customers in the world spend at this restaurant.*3899

*I subtracted 30 from both sides and then I divided by σ, where is my σ, there is my σ right there, the standard deviation.*3908

*I divided by σ and then I multiply by √N on the next line.*3919

*N was the number of customers, there is 25 of them, I’m going to multiply both sides by 5.*3926

*5 is the √25.*3934

*I simplified the numbers here, they have worked out pretty nicely.*3936

*They are rigged to work nicely, simplified down to -0.5 and +1.0.*3940

*We are really looking for the probability between -0.5 and 1.0 for a standard normal variable.*3947

*The way my chart works and some people's chart work a little differently, *3959

*but the way my chart works is it will tell you these tail probabilities.*3959

*It tells you the positive tails but you can work out the negative tails the same way.*3964

*What I will do is, I will look up the two different tails there and subtract them off from 1.*3969

*That will give me this probability in the middle, that I'm really looking for.*3976

*I will confirm those on the next page with the chart, it is 0.3085 and 0.1587.*3982

*Once I look those up, I can drop them back into the calculation and just reduce it down to 0.5328 which is about 53%.*3989

*That is my probability that my restaurant is going to make between $725 and $800 tonight.*3999

*The one missing piece of the puzzle from the previous slide,*4009

*we are still answering example 5 now, is to find those two probabilities.*4012

*We were finding the probabilities of being less than - 0.5 or being bigger than 1.0.*4020

*We actually want to find the probability of being between those cutoffs.*4029

*This chart will tell you the probability of being in the tails.*4033

*The probability of Z being less than -0.5 is the same as being bigger than 0.5.*4039

*It should be in this chart somewhere, here it is 0.5 and 0.00 is 0.3085.*4048

*The probability of Z being bigger than 1.0, here is 1.0, it is 0.1587.*4060

*I took those numbers and I do not think I'm going to rehash all the calculations that I did on the previous side.*4078

*I will just say, you plug those numbers in to the appropriate place on the previous side.*4084

*You can go back and watch it, if you do not remember how it works out.*4089

*It would be good if I can spell previous though.*4094

*We work through some calculations and we came up with a 53% probability *4101

*that this restaurant is going to make between $725 and $800 in their nightly revenue.*4107

*That wraps up example 5, most of the work was done on the previous side.*4118

*You can go back and check it out, just the missing step on the previous side was *4125

*where these two numbers came from, the 0.3085 and 0.1587.*4129

*Where they came from was by looking up 0.5 and 1.0, and then getting these two numbers from my standard normal chart.*4136

*If you do not like using charts, if you have an electronic way of getting probabilities*4146

*for a standard normal distribution, then by all means use that.*4151

*It is definitely something quicker than this slightly archaic method.*4156

*In some probability classes, they are still using charts so I want to show you that way.*4160

*That wraps up this lecture on the central limit theorem and *4166

*that is the last lecture in the probability series here on www.educator.com.*4169

*My name is Will Murray, I have really enjoyed making these lectures.*4174

*I hope you have learned something about probability.*4178

*I hope you have been working through the examples and learning something along with me.*4180

*I thank you very much for sticking with me through these probability lectures.*4186

*I hope you are enjoying your probability class and all your math classes, bye now.*4190

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