For more information, please see full course syllabus of Probability

For more information, please see full course syllabus of Probability

### Normal (Gaussian) 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
- Normal (Gaussian) Distribution
- Formula for the Normal Distribution
- Standard Normal Distribution
- Standard Normal Distribution, Cont.
- Nonstandard Normal Distribution
- Example I: Chance that Standard Normal Variable Will Land Between 1 and 2?
- Example II: What Proportion of the Data Lies within Two Standard Deviations of the Mean?
- Example II: Setting Up the Equation & Graph
- Example II: Solving for z Using the Standard Normal Chart
- Example III: Scores on an Exam
- Example III: Setting Up the Equation & Graph, Part A
- Example III: Setting Up the Equation & Graph, Part B
- Example III: Solving for z Using the Standard Normal Chart, Part A
- Example III: Solving for z Using the Standard Normal Chart, Part B
- Example IV: Temperatures
- Example IV: Setting Up the Equation & Graph
- Example IV: Solving for z Using the Standard Normal Chart
- Example V: Scores on an Exam

- Intro 0:00
- Normal (Gaussian) Distribution 0:35
- Normal (Gaussian) Distribution & The Bell Curve
- Fixed Parameters
- Formula for the Normal Distribution 1:32
- Formula for the Normal Distribution
- Calculating on the Normal Distribution can be Tricky
- Standard Normal Distribution 5:12
- Standard Normal Distribution
- Graphing the Standard Normal Distribution
- Standard Normal Distribution, Cont. 8:30
- Standard Normal Distribution Chart
- Nonstandard Normal Distribution 14:44
- Nonstandard Normal Variable & Associated Standard Normal
- Finding Probabilities for Z
- Example I: Chance that Standard Normal Variable Will Land Between 1 and 2? 16:46
- Example I: Setting Up the Equation & Graph
- Example I: Solving for z Using the Standard Normal Chart
- Example II: What Proportion of the Data Lies within Two Standard Deviations of the Mean? 20:41
- Example II: Setting Up the Equation & Graph
- Example II: Solving for z Using the Standard Normal Chart
- Example III: Scores on an Exam 27:34
- Example III: Setting Up the Equation & Graph, Part A
- Example III: Setting Up the Equation & Graph, Part B
- Example III: Solving for z Using the Standard Normal Chart, Part A
- Example III: Solving for z Using the Standard Normal Chart, Part B
- Example IV: Temperatures 42:54
- Example IV: Setting Up the Equation & Graph
- Example IV: Solving for z Using the Standard Normal Chart
- Example V: Scores on an Exam 48:41
- Example V: Setting Up the Equation & Graph, Part A
- Example V: Setting Up the Equation & Graph, Part B
- Example V: Solving for z Using the Standard Normal Chart, Part A
- Example V: Solving for z Using the Standard Normal Chart, Part B

### Introduction to Probability Online Course

### Transcription: Normal (Gaussian) Distribution

*Hi, welcome back to the probability lectures here on www.educator.com.*0000

*We are working through the continuous distributions right now.*0004

*Our last lecture was on the uniform distribution.*0008

*Today, we are going to talk about what is probably the most important distribution of all,*0011

*it is the famous normal distribution.*0015

*It is also known as the Gaussian distribution, that is a little bit less common.*0018

*Probably, you are more likely to see the words normal distribution for this.*0023

*But, if you do happen to see a Gaussian distribution, it does mean the same thing as the normal distribution.*0028

*The normal distribution or the Gaussian distribution, same thing, is the famous bell curve.*0038

*Let me just draw what I mean by that.*0044

*I think probably everybody seeing a picture of the normal distribution, it is the one that looks like this,*0047

*it is the bell curve.*0054

*There are two fixed parameters that go into every normal distribution.*0057

*Μ is the mean, that is always exactly in the middle, it is a symmetric distribution.*0061

*1/2 the data is to the left of μ and half the data is the right of μ.*0067

*Σ is the standard deviation.*0072

*That is about the distance σ onto the high side of μ and that is approximately the distance of σ to the low side of μ.*0076

*We start out with these two constants μ and σ.*0087

*We will look at the formula for the normal distribution but it is quite complicated,*0091

*and it is rather intimidating to people seeing it for the first time.*0095

*But, it is something that you need to know.*0099

*The important part of the formula here is that, it is basically E ⁻Y².*0101

*That what gives it the basic bell curve shade, the E ⁺Y²*0107

*What we do is we adjust it by all of these different little constant.*0113

*If you are wondering what to focus on here, focus on E ⁺Y², and then look at the correction terms.*0117

*We choose Y to Y – μ here, what that does is, it moves the center of the distribution*0125

*from being centered to 0 over to being centered at the mean μ.*0131

*That is what that adjustment does.*0136

*There is a correction term of 2 σ² which makes it wider or thinner,*0139

*according to the standard deviation that you want to have for your normal distribution.*0144

*Remember that, the total area under any density function always has to be 1.*0150

*There is another correction constant that we put on the normal distribution,*0157

*in order to make the total area under that bell curve B1.*0161

*That correction term is this 1/σ × √ 2π.*0165

*That is just a constant that we multiply on, in order to make the total area come out to be 1.*0171

*The normal distribution does go on infinitely in both directions, let me draw another graph of it.*0178

*Y can take values between -infinity up to infinity.*0185

*It does not have any cutoff, that is different from the uniform distribution which was cut off between two values θ1 and θ2.*0190

*We saw that in the previous lecture.*0197

*The normal distribution does go on forever.*0199

*You can have Y values as small as you want or as big negative as you want, or as big positive as you want.*0203

*It goes on forever in both directions.*0211

*The probability calculating probabilities on the normal distribution turns out to be quite a tricky thing.*0214

*Here is why, normally, with other distributions or even in theory with a normal distribution,*0221

*the way you calculate the probability of a particular range is you integrate the density function over that range.*0230

*That is what you like to do with the normal distribution, you would like to calculate an integral from A to B,*0239

*and then that would give you the probability of the variable falling within that range.*0244

*The problem with the normal distribution is that, this density function cannot be integrated directly.*0250

*There is no way to write down function whose derivative is this density function.*0256

*It is basically the old problem that you can integrate E ⁺Y² directly.*0262

*There is no elementary way to do that.*0269

*That creates all kinds of problems, when you want to calculate probabilities for the normal distribution.*0273

*You cannot just solve things with an integral, the way you can with a lot of other probability problems.*0279

*Let me show you what you do instead.*0286

*Since, you cannot do this integral, there is sort of a way to get around this problem and solve it a totally different way.*0288

*But, it requires that I take some preliminary steps.*0296

*The preliminary steps we are going take are by looking at what is called the standard normal distribution.*0305

*Let us go ahead and talk about the standard normal distribution.*0313

*It is a normal distribution but it is a special one that has the mean as 0 and the standard deviation as σ = 1.*0316

*The notation we use for that is N of 01, that means, mean of 0, that is variance of 1*0326

*but since standard deviation is the square root of variance, the standard deviation also comes out to be 1.*0335

*We often use the variables Z for standard normal distribution, that is kind of enshrined in the folklore.*0342

*You can talk about Z values and that often means the standard normal distribution.*0349

*You will see probably in your own textbook, in your own problems, you will see them talking about variables with Z in there,*0356

*that Z is indicating a standard normal variable.*0364

*It means the mean is 0 and the standard deviation is 1.*0369

*When you are graphing the standard normal variable, the standard normal distribution,*0376

*it is always centered at 0 which makes it a little easier to calculate things.*0381

*It was a little up sided here, I’m not going to worry about that.*0386

*There is a standard normal distribution.*0389

*It is still not possible to calculate probabilities even on a standard normal distribution directly.*0393

*That interval is still impossible even we simplify it by taking μ to be 0 and σ = 1.*0401

*That does not make the integral possible to solve directly.*0408

*What people do, historically, people used charts of values of standard normal distributions.*0412

*These days, your calculator might have a function to calculate probabilities on the standard normal distribution.*0420

*Certainly, computer, algebra systems, things like Mathlab, mathematica, Maxima, some of those computer algebra systems,*0429

*sage is a popular free one online open source system, those will have functions*0439

*to calculate probabilities using the standard normal distribution.*0445

*You can also find standalone applications online.*0451

*You can find a lot of programs online that will calculate the standard normal distribution for you.*0454

*Depending on what is standard in your probability course, you might calculate these different ways.*0461

*What I’m going to do here is, I'm going to use charts of the standard normal distribution and even these charts,*0467

*there are different ways people use these charts.*0475

*I will show you the chart that I have been using and I will show you how to use that.*0478

*It might be a little bit different from the one you have been using,*0482

*you might have to work out how to convert back and forth from my system or your system.*0485

*If you are lucky then you just maybe have some kind of computer program that you just plug in*0490

*the values and it spits out the probabilities for you.*0496

*I will show you how to do that but it does take a little bit of cleverness.*0499

*I will walk you through it and we will see how to do it in some problems.*0506

*Here is a typical standard normal distribution chart.*0512

*I have got the normal curve areas here and then a whole bunch of values.*0517

*Let me show you the way what these numbers represent.*0521

*What you do is you pick your Z value and it will be something like 1.24, for example.*0526

*You pick your Z value is 1.24 and then what this chart will give you is the probability that the variable is bigger than that cut off value.*0536

*The probability that Z is bigger than 1.24.*0548

*The way you read this particular chart is you find 1.2 on the left, here it is, right there.*0557

*And then, you find the second decimal place on the top which is right here, there is 0.04.*0564

*I see that, that probability is 0.1075 according to my chart.*0572

*It is 0.1075 and that is how we calculate the probability that my standard normal variable would be bigger than 1.24.*0582

*Again, your chart, the way you do it in your probability class might be slightly different.*0595

*You might have a chart that is organized differently, it might have the rows and columns switched, or something like that.*0600

*If so, you have to figure out how to make the conversion.*0605

*But, if you chart looks like mine, then this is the way you read it.*0608

*If you are lucky then you can just calculate these things on a calculator and*0611

*you would not even have to use the old fashioned methods.*0615

*This does not show you directly how to calculate probabilities in between two ranges.*0620

*Let me show you how you would calculate that.*0625

*I do not have a lot of space here, I will calculate it.*0629

*Supposed you want to calculate the probability that a variable is between, I'm going to say 0.56 and,*0632

*Let me give myself a little more space for that, I got a little squished in.*0644

*0.56 and 1.24, how would you calculate that using this kind of chart?*0648

*The way we calculate it is, let me draw a little normal variable here.*0659

*What you are trying to find here is the region in between two values there.*0672

*What the chart will tell you is your area bigger than a certain value.*0681

*The way you do this is, you calculate the area bigger than 1.56, that is all that area.*0686

*And then, you subtract off the area that is bigger than 1.24.*0695

*Did I say 1.56, I meant 0.56.*0701

*There is 0.56 and there is 1.24.*0705

*You want to calculate all the area bigger than 0.56, and then, subtract off the area bigger than 1.24.*0710

*That will tell you the area in between them.*0719

*It is the probability that Z is bigger than 0.56 - the probability that Z is bigger than 1.24.*0723

*Each one of those, you can figure out directly from this chart.*0737

*Let me find 0.56 here, here is 0.5 here, and there is 0.06.*0741

*We read across and say it is 0.2877 is 0.2877 - 1.24, we already figure that one out, that is 0.1075.*0749

*We subtract those two, we get 0.18020, that is the probability that we are in between 0.56 and 1.24.*0768

*That is how you would calculate the probability of a range is, you look up two numbers on this table*0784

*and subtract them from each other.*0790

*One other thing to know about this kind of table is, it only gives you positive values.*0794

*If you want to figure out negative value, you would have to kind of figure out*0801

*the corresponding positive value on this table, and flip it.*0805

*For example, if you want to find the probability that Z is bigger than -½ or -0.5, you write that as 0.5.*0810

*If you want to find the probability that Z is bigger than -0.5, you would have to do 1 - the probability*0824

*that Z is less than -0.5, but since it is symmetric, that is the probability that Z is bigger than 0.5.*0832

*That is something you can read off directly from the chart.*0841

*It takes a little bit of getting used to working with this kind of table, and then flipping things around,*0847

*and finding 1- this area/ this area - that area.*0853

*It definitely takes some practice but if you play around with some of these problems, you will start to get the hang of it.*0858

*I got some problems coming up, but before we talk about the problems,*0864

*I got tell you about nonstandard normal distributions because everything here on this chart*0868

*only applies to standard normal distributions.*0874

*I have to show you how to work these things for nonstandard normal distributions, that is the next topic.*0877

*A standard normal means the mean is 0 and the standard deviation is 1.*0889

*Nonstandard just means any other normal distribution, where you do not necessarily*0895

*have a mean being 0 and a standard deviation of 1.*0901

*You just have some mean and some variance which corresponds to some standard deviation.*0904

*The trick for dealing with those is to convert it into a standard normal distribution.*0910

*Here is how you do the conversion.*0917

*You know the mean and the standard deviation, you form this other variable Y – μ/σ.*0919

*It turns out that, that is a standard normal distribution.*0928

*That is why we call it Z is because it does turn out to be a standard normal distribution.*0934

*If you are looking for ranges of Y, if you want to find the probability that Y is between A and B,*0940

*what you do is you convert that into a range for Z.*0947

*If Y is between A and B, if you plug in A and B for Y there, you got Z should be between A – μ/σ and B – μ/σ.*0951

*Z should be between those two ranges.*0962

*The key thing here is that Z is now a standard normal variable.*0965

*We know how to look up probabilities for a standard normal variable.*0971

*We just learned that using the chart or using any of those computational tools that you might have available to you.*0975

*You can look up probabilities for the standard normal variable, and then that will tell you*0982

*the answer for the probabilities for the nonstandard normal variable Y.*0988

*That is kind of the philosophy there.*0993

*It definitely takes some practice to get used to that.*0997

*Let us jump into the problems now and you will see how that works out.*1001

*In the first problem here on the normal distribution, we want to find the probability*1007

*that a standard normal variable will land between 1 and 2.*1011

*Let me draw a little picture of what we are looking at.*1017

*Once we are sure that we understand this, then I will jump to the chart and we will see*1020

*how to look up the numbers on the chart and get an answer.*1027

*This is a standard normal variable, let me go ahead and draw a graph of my standard normal.*1031

*It is always centered at 0, that is what it means to be standard normal is it is centered at 0 and*1037

*has a standard deviation of 1.*1043

*There it is, centered at 0.*1047

*I want the range between, there is 1 and there is 2.*1051

*I’m trying to find this area right here, between 1 and 2.*1057

*I’m trying to find the probability, since it is standard normal, I’m going to call it Z, between 1 and 2.*1064

*That is not something I know how to calculate directly.*1073

*If you remember what my chart will tell me, I will look at the chart is, if I have a value of Z,*1075

*it will tell me the area to the right of that.*1085

*The probability that you are bigger than that particular value.*1088

*The way I can work this out is, this is the probability that we are bigger than 1.*1091

*The probability that Z is bigger than 1 - the probability that Z is bigger than 2.*1099

*We are finding this area in two stages, finding all the area bigger than 1 and then*1110

*subtracting off the area bigger than 2, -the probability that Z is bigger than 2.*1114

*Both of those are things that we will be able to calculate from the standard normal chart.*1124

*I'm going to jump over to the next page where I got a standard normal chart setup.*1132

*Just remember that, we are going to look up the values for Z bigger than 1 and Z bigger than 2.*1137

*We are going to subtract them.*1143

*Here is my standard normal chart, and just remember from the previous page that*1147

*we are trying to find the probability that Z is between 1 and 2.*1152

*We figured out that, we can do that as an area calculation by doing the probability density is bigger than 1 -,*1156

*that is supposed to be bigger than or equal to sign.*1165

*- the probability that Z is bigger than or equal to 2.*1168

*Let us find each one of those on the chart.*1175

*Here is 1.0 and there is 1.00 is 0.1587, 2.00 is 0.0028.*1177

*My probability that we are between 1 and 2, Z is between 1 and 2.*1193

*Z bigger than one is 0.1587 - 0.0028, let us see, that is .1359.*1205

*The probability that we will be between 1 and 2, on the standard normal chart is approximately 0.1359.*1227

*Let us keep moving here.*1241

*In example 2, again, we have a set of data normally distributed.*1243

*What proportion of the data lies within 2 standard deviations of the mean.*1248

*I’m going to draw a little graph here and we will calculate it graphically.*1253

*And then, we will jump to a chart on the next page and we will try to figure out exactly what the numbers turn out to be.*1258

*I will draw my standard normal, by the way, this example does not tell us what the mean is or what the standard deviation is,*1268

*I'm just going to go with the standard normal because it is the easiest one to calculate there.*1282

*There is my standard normal, we are a lopsided there but that will do.*1292

* There is -1, 0, 1, 2, and -2.*1297

*The standard deviation for a standard normal is exactly 1 and the mean is 0.*1306

*There is μ is equal to 0 and the standard deviation is 1.*1312

*2 standard deviations would be, we go 2 down and 2 up from 0.*1319

*We are trying to calculate that area right there.*1324

*We got to do a little bit of graphical cleverness here because remember, what the chart will tell us.*1329

*What the chart will tell me is the probability that we are greater than any particular cutoff value of Z.*1339

*The chart will tell me that area right there for any value of Z that I want to look up.*1349

*That is not what I want, I want this area in between.*1356

*What I notice is that this thing is totally symmetric.*1359

*I can look at that area right there, that tail area.*1364

*If I take two of those tail areas because I’m cutting off two tails, and subtract that from 1,*1370

*that will give me the area that I'm looking for.*1377

*The probability that Z is between -2 and 2, that is what I want, it is equal to 1 -2 of those tail areas.*1380

*But, those tail areas are the same, 2 × the probability that Z is greater than or equal to 2.*1390

*That is something that I can look up fairly quickly.*1399

*I'm going to do that on the next page where I got a nice standard normal chart setup for myself.*1402

*Before I jump to the next page, let me remind you where things came from here.*1408

*We are asked what proportion of data lies within 2 standard deviations of the mean.*1413

*To get 2 standard deviations of the mean, the mean is 0 for standard normal.*1419

*I went up 2 and that 2, because the standard normal is 1, that is why I want to calculate the probability from -2 to 2.*1424

*A clever way to do that is, because I have a chart that will tell me what these tails are, is to take the total area of one,*1437

*that is where that one came from, the total area.*1446

*And subtract off 2 of those tails, that is what I'm subtracting off right there.*1450

*I’m subtracting off 2 of those tails.*1456

*The tail area is the probability that Z is greater than or equal to 2.*1460

*That is what that area is there.*1467

*In order to put some numbers to that, I have to look at a standard normal chart.*1469

*Let me jump to the next slide and go ahead and look at that.*1474

*Here is my standard normal chart, and what I worked out on the previous side is that the probability of Z being between -2*1480

*and 2 is 1 -2 × the probability that Z is bigger than or equal to 2.*1490

*Now that means, I want to find Z bigger than 2 on this chart.*1500

*What these numbers on this chart are telling the me, is the probability that Z is bigger than any particular cut off z.*1506

*In this particular case, z is 200.0, here is 2.0, and here is 0.0.*1514

*Therein is my answer for that probability 0.0228.*1525

*It is 1 - 2 × 0.0228, and I just threw that into my calculator, I did not even bother to calculate the intermediate steps.*1532

*It worked out to be 0.9544, and I did convert that into a percent.*1547

*I wrote that as 95.44%.*1557

*By the way, that is one of the classic values in baby statistics.*1562

*If you took a very introductory level statistics class, maybe in high school, maybe in your first year of college.*1566

*The classic results is that, if you have normally distributed data then 68% of it is within 1 standard deviation of the mean*1575

*and 95% of it is within 2 standard deviations of the mean.*1588

*What we just calculated here is that second number, that 95% of the data,*1593

*that it is actually 95.44% of the data is within 2 standard deviations of the mean.*1597

*Now you know where that classic result from baby statistics comes from.*1607

*If you want to find the 68%, it comes from 1 -2 × that number right there.*1610

*1 -2 × 0.1587 gives you the classic 68% figure of 68% of data within 1 standard deviation of the mean.*1618

*Let us keep moving here. We have been talking about standard normal variables.*1631

*In example 3, we are going to start talking about nonstandard normal variables.*1635

*But remember, the trick for nonstandard normal variables is to convert them back into standard normal variables.*1640

*We will see an example of that with example 3 and you get to practice*1647

*the techniques you have been using for standard normal variables.*1651

*In example 3, we have scores on an exam that are normally distributed.*1655

*The mean is 76 and the variance is 64.*1660

*By the way, that means we have a nonstandard normal variable.*1664

*We no longer have mean 0, we no longer have standard deviation 1.*1667

*We want to find the proportion of scores that are between 72 and 96.*1672

*We want to find, the minimum passing score is 60.*1677

*We want to find how many students will pass or what percentage of students will pass.*1681

*Let me remind you, this is a nonstandard normal variable.*1687

*The trick there, for a nonstandard normal variable is to convert it to a standard normal variable.*1692

*What you do is, you have your nonstandard normal Y and it has some kind of mean and standard deviation,*1699

*you find Y – μ/σ and you call that Z.*1708

*By definition, that Z, and that is a standard known variable.*1712

*You can look up probabilities on Z and then convert them into values for Y.*1717

*In particular, the probability that Y will be within a certain range, you convert that into probabilities for Z.*1724

*That means that Z would be between, if the Y is going to go from A to B, then Z,*1736

*when you plug those values in for Y, A – μ/σ and B – μ/σ.*1743

*That is the trick, and then those are values for a standard normal variable.*1756

*You can look up those values on a standard normal chart or use one of your automated applications*1763

*for calculating values for a standard normal variable.*1771

*In part A here, we want to find the probability that Y is between 72 and 96.*1775

*That is the probability, we want to convert those into probabilities for Z.*1793

*What is our μ and our σ here.*1800

*The μ is 76, that is given as the mean.*1803

*Now, 64 was given as the variance, that means that is σ².*1806

*We know the variance is σ², the standard deviation is always the square root of the variance.*1812

*The standard deviation is 8.*1818

*If Y is between 72 and 96, then Z should be between 72 -76/8 and 96 -76/8.*1824

*I can simplify those, the probability that Z will be between 72 -76 is -4.*1842

*-4/8 is -1/2 or -0.5.*1853

*96 -76 is 20, 20/8 is 2.5.*1862

*I have probabilities for a standard normal variable.*1871

*I just have to find the probability that, that is between -0.5 and 2.5.*1875

*Let me draw a little graph and show you how I plan to figure that out.*1882

*I always seem to make it a little too steep on the positive side.*1891

*Here is my graph of a standard normal variable and we want to find the probability that it is between -0.5 and 2.5.*1894

*I want to find that area in between those 2 bounds.*1908

*It is a little bit tricky, given the way my chart works.*1913

*I think what I’m going to do is figure out each one of those regions to the left and right side of the axis separately.*1917

*This left hand region, I will do ½ because all the region to the left of the Y axis is ½, that is because the total region is 1.*1926

*½ - the probability that Z is bigger than 0.5.*1946

*It is the way I’m going to figure out that left hand region.*1953

*The right hand region is ½ - the probability that Z is bigger than or equal to 2.5.*1956

*I'm going to find this region separately and then I’m going to add them together.*1965

*The reason I use ½ there is because I'm kind of splitting the normal distribution in two.*1969

*There is ½ area to the left and ½ area to the right.*1975

*I’m going to work it from there.*1979

*This is ½ - the probability that Z is bigger than or equal 2.5 + ½ - the probability that Z is bigger than or equal to 2.5.*1984

*I will calculate those.*2001

*Those are both things that I can easily look up on the normal chart.*2002

*I’m just going to leave that and look those up on a normal chart.*2007

*I got a normal chart, I will set up on the next slide.*2009

*And then, I will have the answer to part A.*2015

*For part B, I need the minimum passing score to be 60.*2018

*I want to find the proportion of students that will pass.*2023

*Let us see, this as part A.*2027

*For part B, I want the probability, it is the same as the proportion of students that will pass.*2030

*The probability that Y is bigger than or equal to 60.*2039

*I want to convert that into my standard normal variable.*2044

*I want to write that as a probability on Z.*2049

*Z should be bigger than or equal to.*2054

*Again, I'm going to plug into my values for my conversion from Y into Z.*2058

*Z should be bigger than or equal to 60 – μ is 76/σ was 8.*2067

*That is the probability that Z is bigger than or equal to, 60 -76 is -16/8 is -2.*2080

*Let me graph out what I will be trying to calculate there.*2093

*I want the probability that Z is bigger than or equal to -2.*2100

*That is all of that region right there.*2107

*Again, that is not something that my chart will calculate for me directly.*2112

*I think what I will do is, instead, I will look at the probability that Z is bigger than or equal to 2.*2116

*Z is bigger than or equal to 2 because that is the same as being less than or equal to -2, and then, I will subtract that from 1.*2125

*That will tell me the probability that Z is bigger than or equal to -2.*2133

*It is 1 - the probability that Z is bigger than or equal to -2.*2139

*The reason I'm using 1 - here is because I'm really looking at both sides of the graph together.*2146

*Whereas before, I was only looking at the individual side separately, that is why I used ½ before.*2151

*I think I got this into a form where we can easily look at the normal chart and get an answer for each one of these.*2159

*Let me just recap how I got these, and then we will jump onto the normal chart and we will solve them out.*2169

*In part A, we want Y to be between 72 and 96.*2175

*I plugged that into my conversion formula for Z.*2181

*That means Z should be between A – μ/σ and B-μ/σ.*2184

*My μ and σ came for the problem, there is μ.*2190

*Because of the variance, I knew that 64 was σ², that is how I got σ was8.*2194

*I plugged those in for μ and σ there.*2201

*And then, I simplified that down and I got a range on Z which is a standard normal variable.*2207

*I graph that out here.*2213

*In order to find that area, that is a little bit of a strange area, I just broken up into 2 pieces.*2218

*I broke it up here and I'm going to find each one of those areas separately, by doing ½ - the appropriate tail.*2225

*There is the tail and I do ½ - that tail.*2234

*Here is the tail here and I do ½ - that tail.*2235

*That is where I got those two values there.*2242

*For the second part, part B, I want to find the proportion of students bigger than 60.*2245

*What I'm doing there is, I'm finding Y bigger than 60.*2254

*I put in 60 for Y and then again, I fill in my μ, my Y, and my σ.*2260

*I get a cutoff value of -2 for my standard normal Z.*2267

*In order to find that area, I think the easier way to do it is to flip it around and find the probability that Z is,*2273

*I have a -2 there and what I really want is 2.*2282

*Let me just change that to be a 2.*2287

*1 - the probability that Z is bigger than or equal to 2.*2291

*We will calculate that out and that will give us the answer.*2297

*Once we look at the normal chart on the next page.*2300

*Here is my normal chart, I see that it is in cutoff at 2.0.*2305

*That is unfortunate because we are going to need to go up to 2.5.*2312

*I just have to tell you the values for 2.5.*2315

*But, let me remind you that the answers were from the previous slide.*2319

*You can go back and check, if you do not remember how we got them.*2322

*For part A of this problem, one of the probability that Y was between 72 and 96.*2325

*The conversion values for Z on that, it turned out to be ½ - the probability that Z is bigger than or equal 2.5.*2339

*This is after doing some work on the previous page.*2352

*+ ½ - the probability that Z is bigger than or equal to, I think it was 2.5.*2355

*½ + ½ is 1 - Z bigger than 0.5.*2367

*Here is my 0.50 right here, it comes from 0.5 and 00, 1 - 0.3085.*2372

*2.5 is off the edge of this chart, I'm just going to have to look at a bigger chart which*2382

*I do not have on the screen right here.*2390

*But my bigger chart shows me that somewhere down here would be 2.5.*2392

*The value is a very small, it is 0.0062.*2401

*That is the probability that a standard normal would be bigger than 2.5.*2408

*That is just a decimal that I threw into my calculator.*2416

*Simplify it out, I got .6853, that turns out to be about 68.53%.*2421

*The way you interpret that answer, by the way is, that is the proportion of students on this exam*2438

*that are going to score between 72 and 96 on this exam.*2444

*The answer to part B, remember from the previous slide, we worked it out to be 1 - the probability*2451

*that Z is bigger than or equal to 2.*2458

*Here is 2.00 on here, which is 1 - 0.0228 which is 0.9772 which is 97.72%.*2462

*That is the passing rate on this exam, that is very fortunate for the students involved, almost all of them passed.*2483

*Remember, the original question there was what percentage of students will get about 60.*2491

*Our answer here is that 97, almost 80% of the students will get above 60.*2496

*To recap what we did on this slide, that was the question that we had from before.*2503

*We converted it on the previous slide, into a couple of values from the Z charts.*2512

*The ½ and ½ put together, that is where I got this 1.*2518

*Z being bigger than 0.5 gave me this 0.3085, that is where that came from.*2523

*Z being bigger than 2.5, that is kind of off the chart here.*2530

*I read that off with a bigger chart which I have, it is not on the screen.*2534

*That is where the 0.0062 came from, simplified that down to 68.53%.*2538

*For part B, we are finding the probability that Y is greater than or equal to 60.*2546

*That turned into a Z value, we did that work on the previous slide, you can check it out.*2552

*The Z value of 2 gives us 0.0228 and that corresponds to 97% of students passing this exam.*2558

*A very happy result for the students there.*2570

*In example 4, we are looking at day time high temperatures in Long Beach California.*2576

*It is a very warm, pleasant city to live in.*2581

*The average high temperature is 75, standard deviation is 9.*2583

*Of course, these are in degrees Fahrenheit, if you are watching from somewhere with degrees Celsius,*2589

*you can convert those if you feel so inclined.*2594

*What percentage of the days has high temperatures above 88?*2597

*How many really hot days we will have here?*2601

*Again, this is a nonstandard normal variable.*2604

*The trick to dealing with nonstandard normal variables is to convert them into standard normal variables.*2609

*Let Z, you define Z to be Y – μ/σ.*2618

*That colon means define to be.*2625

*We are defining our Z right there, to be Y – μ/σ.*2630

*The μ and σ are given to us in the problem.*2635

*The mean is μ = 75, the σ is the standard deviation, it is 9.*2637

*You got to be careful and read these problems, and see whether they are talking about standard deviation or variance.*2643

*Because it is a matter of whether you square or take the square root of the number or not.*2649

*In this case, they are saying standard deviation, that is my σ.*2654

*The point is that, we can convert, we want to find how many days have high temperatures over 88.*2660

*How many really hot days do we have or what percentage?*2668

*We want to find the probability that Y is bigger than 88. That converts into Z being bigger than or equal to 88 – μ/σ.*2672

*I just plugged in my value of Y into the definition for Z.*2687

*That is the probability that Z is bigger than or equal to, my μ was 75, my σ is 9.*2692

*That is the probability that Z is bigger than or equal to 88 -75 is 13/9.*2707

*And that is the probability that Z is bigger than or equal to 1.44.*2714

*I just threw 13/9 into a calculator there, actually we can do that without a calculator.*2722

*That means that I want to look up a standard normal variable and find the probability that it is bigger than 1.44.*2730

*That is going to be a fairly quick calculation.*2738

*I will just look it up on the chart, there is 1.44.*2743

*I'm looking for the probability Z being greater than that, I want that.*2747

*That is something I can look at directly on a chart.*2751

*I do not need to do any funny area conversions there.*2754

*I set up my chart on the next page and I will just flip over and look that up.*2758

*But let me a recap this, before I'm bury it forever.*2764

*We got a nonstandard normal distribution here.*2769

*The trick to dealing with nonstandard normal distributions is to convert them*2773

*into standard normal distributions, by doing Y – μ/Σ.*2778

*We get the μ and the σ from the problem, you want to be careful to notice*2782

*whether the problem says variance or standard deviation.*2787

*In this case, it says standard deviation, that is my σ.*2790

*I do not need to take a square root.*2793

*I want to know what percentage of days has temperatures above 88.*2797

* Y should be above 88, that means that Z is above, I plugged in 88 for Y, I get 88– μ/σ.*2801

*88 – 75/9 is 13/9.*2810

*13/9 converts to 1.44.*2814

*I’m about to turn the page and look up 1.44 on a standard normal chart.*2817

*What we figured out on the previous page was that, the probability that Y is bigger than or equal to 88*2826

*was the probability that Z, my standard normal variable is bigger than 1.44.*2833

*That should be somewhere on the chart, here is a 1.4 column, the second decimal place is 4.*2841

*Let me read those together, 1.44 gives me 0.0749.*2848

*0.0749, if we convert that into a percentage, that is 7.49%.*2856

*That is the percentage of days in Long Beach that are going to over 88°, really hot days in Long Beach California.*2866

*Just to recap what we did on this page.*2878

*What we are originally asked is what percentage of days are above 88°?*2881

*From the previous page, we converted that into a standard normal variable.*2887

*Y being bigger than 88 corresponds to Z bigger than 1.44.*2892

*Then, I just looked that up on the chart, got my value, and converted it into a percentage.*2897

*If you are lucky in your class, you will be allowed to use a calculator functions.*2903

*You would not need these charts, or some kind of online application.*2908

*If so, go ahead and use those, and do not worry about using these charts because the charts are a little slower.*2911

*In example 5 here, we have, once again, scores on an exam which are normally distributed.*2922

*The mean is 70 and the variance is 16.*2929

*The first thing to notice here is that, you have to be careful, it says the variance is 16 not the standard deviation.*2934

*That means our σ is actually the square root of the variance which would be 4 here, that mean will be 70.*2941

*We want to figure out what percentages of the scores are between 68 and 78?*2949

*Just like the others, the trick to dealing with these nonstandard normal problems*2955

*is to convert them into standard normal problems.*2959

*And then, you can look up the answers for standard normal probabilities on a chart.*2963

*Again, I'm going to set up a standard normal which is always Y - the μ/ the standard deviation.*2970

*In this case, it is Y – 70/ my standard deviation here is 4.*2980

*The probabilities, if I want to find the probability on Y, I convert that into a probability on Z, that is the probability on Z.*2988

*And then, I just plug in those values of Y into the formula for Z.*2998

*I want to find Y going from A to B.*3009

*That means A- μ/σ and B – μ/σ.*3012

*That is what I have to do to convert a nonstandard normal variable into a standard normal variable.*3022

*In this case, for part A here, we want to find the probability that Y is between 68 and 78.*3030

*Let us convert that into a probability on Z.*3044

*Z should be between, my A and B are 68 and 78.*3049

*I get to 68 – 70/ my standard deviation was 4, and 78 -70/4 which is the probability that Z is between 68 -70 is -2.*3053

*-2/4 is -.5, 78-70 is 8/4 is 2.*3074

*Let me draw a little graph to show how I plan to calculate that.*3086

*That would be a little bit complicated, I need ½ somewhere.*3091

*I can just fill those ½ coming up.*3096

*You want to go from -0.5 to 2.*3099

*I want this area right here.*3105

*As usual, I think I'm going to split that up into two areas.*3110

*The left hand area is, the total area would be ½ - the tail area which is the probability that Z is bigger than ½, bigger than 0.5.*3116

*I’m subtracting off a positive tail because it is symmetric.*3133

*The positive and negative tails are the same area.*3136

*This is ½ - the probability that Z is bigger than or equal to 0.5.*3139

*This right hand area is ½ - the tail, the tail starts at 2 not 2.5.*3150

*½ - the probability that Z is bigger than or equal to 2 + ½ - the probability that Z is bigger than or equal to 2.*3160

*Let me simplify that a little bit.*3173

*Combine the ½ into 1- the probability that Z is bigger than or equal to 0.5 –*3174

*the probability that Z is bigger than or equal to 2.*3185

*I can not move both of those up, as soon as I get a standard normal table for myself.*3191

*I’m going to hang onto those for now and I’m going to look at part B.*3197

*In part B, I want to find the minimum score to be in the top 10% of students.*3201

*Maybe, the top 10% of students will receive honors upon graduation.*3207

*This one is a little different from the previous problem.*3213

*You do not have to think extra hard about this one.*3216

*I want the probability that Z being above some cutoff.*3219

*I do not know what the cutoff is yet, I’m just going to call it z, to be exactly 0.1*3225

*because I want it to be the top 10% of students.*3233

*I want some cutoff and in order to be above that cutoff, you must be in the top 10%.*3236

*That is where I’m getting that 0.1 there.*3243

*I want to figure out what z would have to be, solved for z.*3245

*I have to use that normal chart to figure that out.*3253

*And then, I want to figure out what the cutoff score is for that because that is really going to be the Z value.*3258

*Then, I will get Z is equal to Y – μ/σ.*3266

*Once I figure out what Z is, I can solve for Y.*3272

*I will get Z × σ is equal to Y – μ.*3276

*Y would be equal to Z σ, I will write it as σ Z.*3284

*Σ Z + μ, those solve for Y.*3291

*I can fill the values of σ and μ, I know what those are right now.*3298

*Σ is 4 and my μ was 70, 4Z + 70.*3302

*What I will do is, I will solve for Z.*3312

*I have to do that by looking at the normal chart on the next page.*3314

*Then, I will figure out my cutoff value for Y, after I figure out what Z is.*3316

*That will be the minimum exam score to get you in the honors category.*3324

*Before I go ahead and jump to the Z charts on the next page, let me show you how I calculated everything here.*3331

*We want to find the probability that Y was between 68 and 78.*3336

*Y is a nonstandard normal variable.*3341

*I converted into a standard normal variable here, Y – μ/σ.*3345

*My μ is given by the problem here.*3350

*This σ is variance, the standard deviation is the square root of the variance.*3353

*The standard deviation is √ 16, that is why I used 4 for the standard deviation*3361

*If I have a range of values for Y, here is how I convert it into a range of values for Z.*3367

*I just plug in the A and B for Y into the equation for Z.*3374

*I plugged those in, the A and B are 68 to 78, in this case.*3380

*I got a range for Z, and then, I looked at that on my graph.*3384

*I’m going to split that up and calculate each one of those ranges separately.*3391

*Each one is ½ - the tail value because there is always ½ of the area on the left and ½ of the area on the right.*3395

*I'm really calculating ½ - these two tail values.*3405

*Those are the two tail values right there that I'm going to have to look up on the normal chart, on the next page.*3410

*That is how I got this answer that I’m filling the details on the next page.*3416

*For part B, I want to calculate the minimum score to be in the top 10% of students.*3423

*I want to figure out what cutoff on the Z chart will give me a probability of 10% of being above that score.*3429

*I will solve that for Z on the next page.*3440

*Once I find a Z cut off, I will convert that back into a Y.*3446

*I solved this back into Y using μ and σ.*3451

*I figure out a Y cut off that will give me the exam score required to put you in the top 10% of students.*3455

*Let us jump forward to next page.*3464

*Here is my Z chart, and let me remind you what I calculated on the previous page.*3467

*In part A, we wanted the exam scores to be between 68 and 78.*3473

*At some cost of work on the previous page, we converted that into some numbers that we can look up on the Z chart.*3482

*1 - the probability that Z is bigger than 0.5 - the probability that Z was bigger than 2.*3493

*I think both of those are here on the chart, 0.5 is right there, 0.3085.*3505

*.2 is .0228, this is 1 - 0.3085 - 0.0228.*3514

*I reduced those and simplify those on my calculator, I just round it to he nearest percentage.*3533

*I got 0.67 rounded to 67%, that is the percentage of students that will score between 68 and 78 on this exam.*3541

*In the second part of the problem there, we want to find the minimum score above which only 10% of students score.*3558

*The first thing was to solve for cutoff value of Z.*3577

*I want to solve, find z.*3583

*It says that the probability of being bigger than that is 0.10, that is from the 10% given in the problem.*3592

*Let us see, I'm looking for 0.10 in the chart.*3599

*If I look through here, they are getting close to 0.10.*3603

*It is really close to that number there, that is 1.2 and 0.08 is 1.28, that is my Z value.*3607

*Let me write that as, my z is 1.28.*3623

*Exactly 10% of the students or very close to 10% of the students are above a Z score of 1.28.*3630

*But now, I have to convert that back into a y score on the exam.*3640

*My Y, I work this out on the previous page was, 4Z + 70 that is 4 × 1.28 + 70.*3644

*If you calculate that out, it comes out to be very close to 75, just slightly above 75.*3662

*What that means is that, if you score above 75 on this exam then you will be in the top 10% of students.*3671

*If you take that Y value and convert it into a Z score, you get a Z score of 1.28 which corresponds to the probability of 0.10.*3683

*Let me recap here.*3694

*A lot of this work was done on the previous page, converting the values for Y into standard normal values.*3696

*We drew some pictures to figure out that we can calculate the standard normal values,*3707

*in terms of these two cutoffs.*3711

*On this page, we just looked up 0.3085 here and 0.0028 here.*3713

*Those corresponded to 0.5 and 2.0.*3721

*And then, I just did the arithmetic and simplified that down to an approximate*3725

*67% of the students scoring between 68 and 78.*3730

*In part B, we want to find what score you had to get to be in the top 10%.*3734

*This is really sort of a reverse engineering problem.*3741

*We started out with the probability of 0.10.*3745

*I had to find that in the chart and I found it right here, very close to 1.28.*3749

*There is the 1.2, there is the 8.*3755

*That is where I got the Z value of 1.28.*3758

*Again, I work this out, I solve that backwards with a σ and a μ from the previous page.*3763

*To solve for Y in terms of Z, I plugged the Z value of 1.28 in, plugged in my σ and μ, and I got a Y score of 75.*3775

*That means you have to score 75 or better on this exam, in order to land within the top 10% of students.*3786

*If you are an honor student, you want to make sure you get those honors by being in the top 10%.*3794

*You better score a 75 or better on this exam.*3800

*That is the last example, that wraps up this lecture on the normal distribution, also known as the Gaussian distribution.*3806

*This is part of the probability lecture series here on www.educator.com.*3815

*The next lecture is on the gamma distribution which also includes the exponential distribution, and the Chi square distribution.*3819

*If you are interested in many of those, I hope you will stick around.*3828

*My name is Will Murray, thank you very much for watching, bye.*3831

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