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Normal (Gaussian) Distribution

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Normal (Gaussian) Distribution

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  1. 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 III: Scores on an Exam
    • Example IV: Temperatures
    • 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

    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 ⁺Y²0107

    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 distribution0125

    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 functions0439

    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 and0611

    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 chart0868

    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 then1110

    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 that1147

    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 applications1763

    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 which2382

    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 probability2451

    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 notice2782

    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 882826

    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 problems2955

    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 0.13225

    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 deviation3361

    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 approximate3725

    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