WEBVTT mathematics/probability/murray
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Hi, welcome back to the probability lectures here on www.educator.com.
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We are working through the continuous distributions right now.
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Our last lecture was on the uniform distribution.
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Today, we are going to talk about what is probably the most important distribution of all,
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it is the famous normal distribution.
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It is also known as the Gaussian distribution, that is a little bit less common.
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Probably, you are more likely to see the words normal distribution for this.
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But, if you do happen to see a Gaussian distribution, it does mean the same thing as the normal distribution.
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The normal distribution or the Gaussian distribution, same thing, is the famous bell curve.
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Let me just draw what I mean by that.
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I think probably everybody seeing a picture of the normal distribution, it is the one that looks like this,
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it is the bell curve.
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There are two fixed parameters that go into every normal distribution.
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Μ is the mean, that is always exactly in the middle, it is a symmetric distribution.
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1/2 the data is to the left of μ and half the data is the right of μ.
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Σ is the standard deviation.
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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.
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We start out with these two constants μ and σ.
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We will look at the formula for the normal distribution but it is quite complicated,
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and it is rather intimidating to people seeing it for the first time.
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But, it is something that you need to know.
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The important part of the formula here is that, it is basically E ⁻Y².
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That what gives it the basic bell curve shade, the E ⁺Y²
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What we do is we adjust it by all of these different little constant.
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If you are wondering what to focus on here, focus on E ⁺Y², and then look at the correction terms.
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We choose Y to Y – μ here, what that does is, it moves the center of the distribution
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from being centered to 0 over to being centered at the mean μ.
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That is what that adjustment does.
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There is a correction term of 2 σ² which makes it wider or thinner,
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according to the standard deviation that you want to have for your normal distribution.
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Remember that, the total area under any density function always has to be 1.
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There is another correction constant that we put on the normal distribution,
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in order to make the total area under that bell curve B1.
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That correction term is this 1/σ × √ 2π.
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That is just a constant that we multiply on, in order to make the total area come out to be 1.
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The normal distribution does go on infinitely in both directions, let me draw another graph of it.
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Y can take values between -infinity up to infinity.
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It does not have any cutoff, that is different from the uniform distribution which was cut off between two values θ1 and θ2.
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We saw that in the previous lecture.
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The normal distribution does go on forever.
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You can have Y values as small as you want or as big negative as you want, or as big positive as you want.
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It goes on forever in both directions.
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The probability calculating probabilities on the normal distribution turns out to be quite a tricky thing.
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Here is why, normally, with other distributions or even in theory with a normal distribution,
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the way you calculate the probability of a particular range is you integrate the density function over that range.
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That is what you like to do with the normal distribution, you would like to calculate an integral from A to B,
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and then that would give you the probability of the variable falling within that range.
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The problem with the normal distribution is that, this density function cannot be integrated directly.
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There is no way to write down function whose derivative is this density function.
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It is basically the old problem that you can integrate E ⁺Y² directly.
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There is no elementary way to do that.
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That creates all kinds of problems, when you want to calculate probabilities for the normal distribution.
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You cannot just solve things with an integral, the way you can with a lot of other probability problems.
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Let me show you what you do instead.
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Since, you cannot do this integral, there is sort of a way to get around this problem and solve it a totally different way.
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But, it requires that I take some preliminary steps.
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The preliminary steps we are going take are by looking at what is called the standard normal distribution.
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Let us go ahead and talk about the standard normal distribution.
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It is a normal distribution but it is a special one that has the mean as 0 and the standard deviation as σ = 1.
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The notation we use for that is N of 01, that means, mean of 0, that is variance of 1
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but since standard deviation is the square root of variance, the standard deviation also comes out to be 1.
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We often use the variables Z for standard normal distribution, that is kind of enshrined in the folklore.
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You can talk about Z values and that often means the standard normal distribution.
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You will see probably in your own textbook, in your own problems, you will see them talking about variables with Z in there,
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that Z is indicating a standard normal variable.
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It means the mean is 0 and the standard deviation is 1.
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When you are graphing the standard normal variable, the standard normal distribution,
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it is always centered at 0 which makes it a little easier to calculate things.
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It was a little up sided here, I’m not going to worry about that.
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There is a standard normal distribution.
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It is still not possible to calculate probabilities even on a standard normal distribution directly.
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That interval is still impossible even we simplify it by taking μ to be 0 and σ = 1.
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That does not make the integral possible to solve directly.
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What people do, historically, people used charts of values of standard normal distributions.
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These days, your calculator might have a function to calculate probabilities on the standard normal distribution.
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Certainly, computer, algebra systems, things like Mathlab, mathematica, Maxima, some of those computer algebra systems,
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sage is a popular free one online open source system, those will have functions
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to calculate probabilities using the standard normal distribution.
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You can also find standalone applications online.
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You can find a lot of programs online that will calculate the standard normal distribution for you.
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Depending on what is standard in your probability course, you might calculate these different ways.
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What I’m going to do here is, I'm going to use charts of the standard normal distribution and even these charts,
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there are different ways people use these charts.
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I will show you the chart that I have been using and I will show you how to use that.
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It might be a little bit different from the one you have been using,
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you might have to work out how to convert back and forth from my system or your system.
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If you are lucky then you just maybe have some kind of computer program that you just plug in
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the values and it spits out the probabilities for you.
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I will show you how to do that but it does take a little bit of cleverness.
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I will walk you through it and we will see how to do it in some problems.
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Here is a typical standard normal distribution chart.
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I have got the normal curve areas here and then a whole bunch of values.
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Let me show you the way what these numbers represent.
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What you do is you pick your Z value and it will be something like 1.24, for example.
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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.
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The probability that Z is bigger than 1.24.
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The way you read this particular chart is you find 1.2 on the left, here it is, right there.
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And then, you find the second decimal place on the top which is right here, there is 0.04.
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I see that, that probability is 0.1075 according to my chart.
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It is 0.1075 and that is how we calculate the probability that my standard normal variable would be bigger than 1.24.
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Again, your chart, the way you do it in your probability class might be slightly different.
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You might have a chart that is organized differently, it might have the rows and columns switched, or something like that.
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If so, you have to figure out how to make the conversion.
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But, if you chart looks like mine, then this is the way you read it.
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If you are lucky then you can just calculate these things on a calculator and
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you would not even have to use the old fashioned methods.
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This does not show you directly how to calculate probabilities in between two ranges.
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Let me show you how you would calculate that.
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I do not have a lot of space here, I will calculate it.
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Supposed you want to calculate the probability that a variable is between, I'm going to say 0.56 and,
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Let me give myself a little more space for that, I got a little squished in.
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0.56 and 1.24, how would you calculate that using this kind of chart?
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The way we calculate it is, let me draw a little normal variable here.
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What you are trying to find here is the region in between two values there.
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What the chart will tell you is your area bigger than a certain value.
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The way you do this is, you calculate the area bigger than 1.56, that is all that area.
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And then, you subtract off the area that is bigger than 1.24.
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Did I say 1.56, I meant 0.56.
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There is 0.56 and there is 1.24.
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You want to calculate all the area bigger than 0.56, and then, subtract off the area bigger than 1.24.
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That will tell you the area in between them.
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It is the probability that Z is bigger than 0.56 - the probability that Z is bigger than 1.24.
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Each one of those, you can figure out directly from this chart.
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Let me find 0.56 here, here is 0.5 here, and there is 0.06.
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We read across and say it is 0.2877 is 0.2877 - 1.24, we already figure that one out, that is 0.1075.
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We subtract those two, we get 0.18020, that is the probability that we are in between 0.56 and 1.24.
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That is how you would calculate the probability of a range is, you look up two numbers on this table
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and subtract them from each other.
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One other thing to know about this kind of table is, it only gives you positive values.
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If you want to figure out negative value, you would have to kind of figure out
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the corresponding positive value on this table, and flip it.
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For example, if you want to find the probability that Z is bigger than -½ or -0.5, you write that as 0.5.
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If you want to find the probability that Z is bigger than -0.5, you would have to do 1 - the probability
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that Z is less than -0.5, but since it is symmetric, that is the probability that Z is bigger than 0.5.
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That is something you can read off directly from the chart.
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It takes a little bit of getting used to working with this kind of table, and then flipping things around,
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and finding 1- this area/ this area - that area.
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It definitely takes some practice but if you play around with some of these problems, you will start to get the hang of it.
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I got some problems coming up, but before we talk about the problems,
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I got tell you about nonstandard normal distributions because everything here on this chart
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only applies to standard normal distributions.
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I have to show you how to work these things for nonstandard normal distributions, that is the next topic.
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A standard normal means the mean is 0 and the standard deviation is 1.
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Nonstandard just means any other normal distribution, where you do not necessarily
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have a mean being 0 and a standard deviation of 1.
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You just have some mean and some variance which corresponds to some standard deviation.
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The trick for dealing with those is to convert it into a standard normal distribution.
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Here is how you do the conversion.
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You know the mean and the standard deviation, you form this other variable Y – μ/σ.
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It turns out that, that is a standard normal distribution.
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That is why we call it Z is because it does turn out to be a standard normal distribution.
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If you are looking for ranges of Y, if you want to find the probability that Y is between A and B,
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what you do is you convert that into a range for Z.
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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.
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Z should be between those two ranges.
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The key thing here is that Z is now a standard normal variable.
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We know how to look up probabilities for a standard normal variable.
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We just learned that using the chart or using any of those computational tools that you might have available to you.
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You can look up probabilities for the standard normal variable, and then that will tell you
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the answer for the probabilities for the nonstandard normal variable Y.
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That is kind of the philosophy there.
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It definitely takes some practice to get used to that.
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Let us jump into the problems now and you will see how that works out.
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In the first problem here on the normal distribution, we want to find the probability
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that a standard normal variable will land between 1 and 2.
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Let me draw a little picture of what we are looking at.
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Once we are sure that we understand this, then I will jump to the chart and we will see
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how to look up the numbers on the chart and get an answer.
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This is a standard normal variable, let me go ahead and draw a graph of my standard normal.
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It is always centered at 0, that is what it means to be standard normal is it is centered at 0 and
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has a standard deviation of 1.
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There it is, centered at 0.
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I want the range between, there is 1 and there is 2.
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I’m trying to find this area right here, between 1 and 2.
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I’m trying to find the probability, since it is standard normal, I’m going to call it Z, between 1 and 2.
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That is not something I know how to calculate directly.
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If you remember what my chart will tell me, I will look at the chart is, if I have a value of Z,
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it will tell me the area to the right of that.
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The probability that you are bigger than that particular value.
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The way I can work this out is, this is the probability that we are bigger than 1.
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The probability that Z is bigger than 1 - the probability that Z is bigger than 2.
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We are finding this area in two stages, finding all the area bigger than 1 and then
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subtracting off the area bigger than 2, -the probability that Z is bigger than 2.
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Both of those are things that we will be able to calculate from the standard normal chart.
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I'm going to jump over to the next page where I got a standard normal chart setup.
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Just remember that, we are going to look up the values for Z bigger than 1 and Z bigger than 2.
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We are going to subtract them.
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Here is my standard normal chart, and just remember from the previous page that
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we are trying to find the probability that Z is between 1 and 2.
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We figured out that, we can do that as an area calculation by doing the probability density is bigger than 1 -,
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that is supposed to be bigger than or equal to sign.
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- the probability that Z is bigger than or equal to 2.
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Let us find each one of those on the chart.
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Here is 1.0 and there is 1.00 is 0.1587, 2.00 is 0.0028.
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My probability that we are between 1 and 2, Z is between 1 and 2.
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Z bigger than one is 0.1587 - 0.0028, let us see, that is .1359.
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The probability that we will be between 1 and 2, on the standard normal chart is approximately 0.1359.
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Let us keep moving here.
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In example 2, again, we have a set of data normally distributed.
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What proportion of the data lies within 2 standard deviations of the mean.
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I’m going to draw a little graph here and we will calculate it graphically.
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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.
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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,
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I'm just going to go with the standard normal because it is the easiest one to calculate there.
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There is my standard normal, we are a lopsided there but that will do.
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There is -1, 0, 1, 2, and -2.
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The standard deviation for a standard normal is exactly 1 and the mean is 0.
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There is μ is equal to 0 and the standard deviation is 1.
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2 standard deviations would be, we go 2 down and 2 up from 0.
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We are trying to calculate that area right there.
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We got to do a little bit of graphical cleverness here because remember, what the chart will tell us.
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What the chart will tell me is the probability that we are greater than any particular cutoff value of Z.
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The chart will tell me that area right there for any value of Z that I want to look up.
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That is not what I want, I want this area in between.
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What I notice is that this thing is totally symmetric.
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I can look at that area right there, that tail area.
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If I take two of those tail areas because I’m cutting off two tails, and subtract that from 1,
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that will give me the area that I'm looking for.
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The probability that Z is between -2 and 2, that is what I want, it is equal to 1 -2 of those tail areas.
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But, those tail areas are the same, 2 × the probability that Z is greater than or equal to 2.
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That is something that I can look up fairly quickly.
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I'm going to do that on the next page where I got a nice standard normal chart setup for myself.
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Before I jump to the next page, let me remind you where things came from here.
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We are asked what proportion of data lies within 2 standard deviations of the mean.
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To get 2 standard deviations of the mean, the mean is 0 for standard normal.
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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.
00:23:57.600 --> 00:24:06.100
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,
00:24:06.100 --> 00:24:10.300
that is where that one came from, the total area.
00:24:10.300 --> 00:24:16.600
And subtract off 2 of those tails, that is what I'm subtracting off right there.
00:24:16.600 --> 00:24:19.700
I’m subtracting off 2 of those tails.
00:24:19.700 --> 00:24:27.000
The tail area is the probability that Z is greater than or equal to 2.
00:24:27.000 --> 00:24:29.100
That is what that area is there.
00:24:29.100 --> 00:24:33.900
In order to put some numbers to that, I have to look at a standard normal chart.
00:24:33.900 --> 00:24:40.400
Let me jump to the next slide and go ahead and look at that.
00:24:40.400 --> 00:24:50.400
Here is my standard normal chart, and what I worked out on the previous side is that the probability of Z being between -2
00:24:50.400 --> 00:24:59.800
and 2 is 1 -2 × the probability that Z is bigger than or equal to 2.
00:24:59.800 --> 00:25:05.700
Now that means, I want to find Z bigger than 2 on this chart.
00:25:05.700 --> 00:25:14.600
What these numbers on this chart are telling the me, is the probability that Z is bigger than any particular cut off z.
00:25:14.600 --> 00:25:25.600
In this particular case, z is 200.0, here is 2.0, and here is 0.0.
00:25:25.600 --> 00:25:31.800
Therein is my answer for that probability 0.0228.
00:25:31.800 --> 00:25:47.100
It is 1 - 2 × 0.0228, and I just threw that into my calculator, I did not even bother to calculate the intermediate steps.
00:25:47.100 --> 00:25:56.700
It worked out to be 0.9544, and I did convert that into a percent.
00:25:56.700 --> 00:26:02.100
I wrote that as 95.44%.
00:26:02.100 --> 00:26:06.500
By the way, that is one of the classic values in baby statistics.
00:26:06.500 --> 00:26:15.200
If you took a very introductory level statistics class, maybe in high school, maybe in your first year of college.
00:26:15.200 --> 00:26:28.100
The classic results is that, if you have normally distributed data then 68% of it is within 1 standard deviation of the mean
00:26:28.100 --> 00:26:33.200
and 95% of it is within 2 standard deviations of the mean.
00:26:33.200 --> 00:26:37.500
What we just calculated here is that second number, that 95% of the data,
00:26:37.500 --> 00:26:47.000
that it is actually 95.44% of the data is within 2 standard deviations of the mean.
00:26:47.000 --> 00:26:50.600
Now you know where that classic result from baby statistics comes from.
00:26:50.600 --> 00:26:58.200
If you want to find the 68%, it comes from 1 -2 × that number right there.
00:26:58.200 --> 00:27:11.100
1 -2 × 0.1587 gives you the classic 68% figure of 68% of data within 1 standard deviation of the mean.
00:27:11.100 --> 00:27:15.300
Let us keep moving here. We have been talking about standard normal variables.
00:27:15.300 --> 00:27:20.400
In example 3, we are going to start talking about nonstandard normal variables.
00:27:20.400 --> 00:27:26.900
But remember, the trick for nonstandard normal variables is to convert them back into standard normal variables.
00:27:26.900 --> 00:27:31.500
We will see an example of that with example 3 and you get to practice
00:27:31.500 --> 00:27:35.300
the techniques you have been using for standard normal variables.
00:27:35.300 --> 00:27:39.700
In example 3, we have scores on an exam that are normally distributed.
00:27:39.700 --> 00:27:43.700
The mean is 76 and the variance is 64.
00:27:43.700 --> 00:27:47.500
By the way, that means we have a nonstandard normal variable.
00:27:47.500 --> 00:27:52.000
We no longer have mean 0, we no longer have standard deviation 1.
00:27:52.000 --> 00:27:57.300
We want to find the proportion of scores that are between 72 and 96.
00:27:57.300 --> 00:28:00.700
We want to find, the minimum passing score is 60.
00:28:00.700 --> 00:28:07.600
We want to find how many students will pass or what percentage of students will pass.
00:28:07.600 --> 00:28:12.000
Let me remind you, this is a nonstandard normal variable.
00:28:12.000 --> 00:28:19.200
The trick there, for a nonstandard normal variable is to convert it to a standard normal variable.
00:28:19.200 --> 00:28:28.100
What you do is, you have your nonstandard normal Y and it has some kind of mean and standard deviation,
00:28:28.100 --> 00:28:32.300
you find Y – μ/σ and you call that Z.
00:28:32.300 --> 00:28:37.000
By definition, that Z, and that is a standard known variable.
00:28:37.000 --> 00:28:44.600
You can look up probabilities on Z and then convert them into values for Y.
00:28:44.600 --> 00:28:55.900
In particular, the probability that Y will be within a certain range, you convert that into probabilities for Z.
00:28:55.900 --> 00:29:02.800
That means that Z would be between, if the Y is going to go from A to B, then Z,
00:29:02.800 --> 00:29:16.400
when you plug those values in for Y, A – μ/σ and B – μ/σ.
00:29:16.400 --> 00:29:23.000
That is the trick, and then those are values for a standard normal variable.
00:29:23.000 --> 00:29:30.800
You can look up those values on a standard normal chart or use one of your automated applications
00:29:30.800 --> 00:29:35.000
for calculating values for a standard normal variable.
00:29:35.000 --> 00:29:52.900
In part A here, we want to find the probability that Y is between 72 and 96.
00:29:52.900 --> 00:30:00.000
That is the probability, we want to convert those into probabilities for Z.
00:30:00.000 --> 00:30:03.300
What is our μ and our σ here.
00:30:03.300 --> 00:30:06.400
The μ is 76, that is given as the mean.
00:30:06.400 --> 00:30:12.200
Now, 64 was given as the variance, that means that is σ².
00:30:12.200 --> 00:30:18.200
We know the variance is σ², the standard deviation is always the square root of the variance.
00:30:18.200 --> 00:30:24.200
The standard deviation is 8.
00:30:24.200 --> 00:30:42.200
If Y is between 72 and 96, then Z should be between 72 -76/8 and 96 -76/8.
00:30:42.200 --> 00:30:52.900
I can simplify those, the probability that Z will be between 72 -76 is -4.
00:30:52.900 --> 00:31:02.500
-4/8 is -1/2 or -0.5.
00:31:02.500 --> 00:31:11.400
96 -76 is 20, 20/8 is 2.5.
00:31:11.400 --> 00:31:15.600
I have probabilities for a standard normal variable.
00:31:15.600 --> 00:31:22.500
I just have to find the probability that, that is between -0.5 and 2.5.
00:31:22.500 --> 00:31:31.200
Let me draw a little graph and show you how I plan to figure that out.
00:31:31.200 --> 00:31:34.400
I always seem to make it a little too steep on the positive side.
00:31:34.400 --> 00:31:48.500
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.
00:31:48.500 --> 00:31:53.400
I want to find that area in between those 2 bounds.
00:31:53.400 --> 00:31:57.200
It is a little bit tricky, given the way my chart works.
00:31:57.200 --> 00:32:05.800
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.
00:32:05.800 --> 00:32:26.100
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.
00:32:26.100 --> 00:32:32.800
½ - the probability that Z is bigger than 0.5.
00:32:32.800 --> 00:32:35.900
It is the way I’m going to figure out that left hand region.
00:32:35.900 --> 00:32:45.300
The right hand region is ½ - the probability that Z is bigger than or equal to 2.5.
00:32:45.300 --> 00:32:49.300
I'm going to find this region separately and then I’m going to add them together.
00:32:49.300 --> 00:32:55.100
The reason I use ½ there is because I'm kind of splitting the normal distribution in two.
00:32:55.100 --> 00:32:59.500
There is ½ area to the left and ½ area to the right.
00:32:59.500 --> 00:33:04.000
I’m going to work it from there.
00:33:04.000 --> 00:33:21.600
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.
00:33:21.600 --> 00:33:22.600
I will calculate those.
00:33:22.600 --> 00:33:26.900
Those are both things that I can easily look up on the normal chart.
00:33:26.900 --> 00:33:29.600
I’m just going to leave that and look those up on a normal chart.
00:33:29.600 --> 00:33:35.400
I got a normal chart, I will set up on the next slide.
00:33:35.400 --> 00:33:37.800
And then, I will have the answer to part A.
00:33:37.800 --> 00:33:43.300
For part B, I need the minimum passing score to be 60.
00:33:43.300 --> 00:33:47.300
I want to find the proportion of students that will pass.
00:33:47.300 --> 00:33:50.000
Let us see, this as part A.
00:33:50.000 --> 00:33:59.100
For part B, I want the probability, it is the same as the proportion of students that will pass.
00:33:59.100 --> 00:34:03.800
The probability that Y is bigger than or equal to 60.
00:34:03.800 --> 00:34:08.900
I want to convert that into my standard normal variable.
00:34:08.900 --> 00:34:14.000
I want to write that as a probability on Z.
00:34:14.000 --> 00:34:18.400
Z should be bigger than or equal to.
00:34:18.400 --> 00:34:26.700
Again, I'm going to plug into my values for my conversion from Y into Z.
00:34:26.700 --> 00:34:40.000
Z should be bigger than or equal to 60 – μ is 76/σ was 8.
00:34:40.000 --> 00:34:53.400
That is the probability that Z is bigger than or equal to, 60 -76 is -16/8 is -2.
00:34:53.400 --> 00:34:59.900
Let me graph out what I will be trying to calculate there.
00:34:59.900 --> 00:35:06.800
I want the probability that Z is bigger than or equal to -2.
00:35:06.800 --> 00:35:11.700
That is all of that region right there.
00:35:11.700 --> 00:35:15.700
Again, that is not something that my chart will calculate for me directly.
00:35:15.700 --> 00:35:25.200
I think what I will do is, instead, I will look at the probability that Z is bigger than or equal to 2.
00:35:25.200 --> 00:35:33.000
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.
00:35:33.000 --> 00:35:38.800
That will tell me the probability that Z is bigger than or equal to -2.
00:35:38.800 --> 00:35:45.700
It is 1 - the probability that Z is bigger than or equal to -2.
00:35:45.700 --> 00:35:51.500
The reason I'm using 1 - here is because I'm really looking at both sides of the graph together.
00:35:51.500 --> 00:35:59.500
Whereas before, I was only looking at the individual side separately, that is why I used ½ before.
00:35:59.500 --> 00:36:09.100
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.
00:36:09.100 --> 00:36:14.900
Let me just recap how I got these, and then we will jump onto the normal chart and we will solve them out.
00:36:14.900 --> 00:36:20.700
In part A, we want Y to be between 72 and 96.
00:36:20.700 --> 00:36:24.100
I plugged that into my conversion formula for Z.
00:36:24.100 --> 00:36:30.600
That means Z should be between A – μ/σ and B-μ/σ.
00:36:30.600 --> 00:36:34.100
My μ and σ came for the problem, there is μ.
00:36:34.100 --> 00:36:40.700
Because of the variance, I knew that 64 was σ², that is how I got σ was8.
00:36:40.700 --> 00:36:47.000
I plugged those in for μ and σ there.
00:36:47.000 --> 00:36:53.400
And then, I simplified that down and I got a range on Z which is a standard normal variable.
00:36:53.400 --> 00:36:57.900
I graph that out here.
00:36:57.900 --> 00:37:05.500
In order to find that area, that is a little bit of a strange area, I just broken up into 2 pieces.
00:37:05.500 --> 00:37:14.400
I broke it up here and I'm going to find each one of those areas separately, by doing ½ - the appropriate tail.
00:37:14.400 --> 00:37:15.200
There is the tail and I do ½ - that tail.
00:37:15.200 --> 00:37:21.700
Here is the tail here and I do ½ - that tail.
00:37:21.700 --> 00:37:25.100
That is where I got those two values there.
00:37:25.100 --> 00:37:33.800
For the second part, part B, I want to find the proportion of students bigger than 60.
00:37:33.800 --> 00:37:39.800
What I'm doing there is, I'm finding Y bigger than 60.
00:37:39.800 --> 00:37:47.600
I put in 60 for Y and then again, I fill in my μ, my Y, and my σ.
00:37:47.600 --> 00:37:53.600
I get a cutoff value of -2 for my standard normal Z.
00:37:53.600 --> 00:38:02.300
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,
00:38:02.300 --> 00:38:07.400
I have a -2 there and what I really want is 2.
00:38:07.400 --> 00:38:11.400
Let me just change that to be a 2.
00:38:11.400 --> 00:38:17.000
1 - the probability that Z is bigger than or equal to 2.
00:38:17.000 --> 00:38:20.600
We will calculate that out and that will give us the answer.
00:38:20.600 --> 00:38:25.500
Once we look at the normal chart on the next page.
00:38:25.500 --> 00:38:32.600
Here is my normal chart, I see that it is in cutoff at 2.0.
00:38:32.600 --> 00:38:35.300
That is unfortunate because we are going to need to go up to 2.5.
00:38:35.300 --> 00:38:39.000
I just have to tell you the values for 2.5.
00:38:39.000 --> 00:38:41.700
But, let me remind you that the answers were from the previous slide.
00:38:41.700 --> 00:38:45.000
You can go back and check, if you do not remember how we got them.
00:38:45.000 --> 00:38:59.500
For part A of this problem, one of the probability that Y was between 72 and 96.
00:38:59.500 --> 00:39:12.000
The conversion values for Z on that, it turned out to be ½ - the probability that Z is bigger than or equal 2.5.
00:39:12.000 --> 00:39:15.100
This is after doing some work on the previous page.
00:39:15.100 --> 00:39:27.100
+ ½ - the probability that Z is bigger than or equal to, I think it was 2.5.
00:39:27.100 --> 00:39:32.000
½ + ½ is 1 - Z bigger than 0.5.
00:39:32.000 --> 00:39:42.300
Here is my 0.50 right here, it comes from 0.5 and 00, 1 - 0.3085.
00:39:42.300 --> 00:39:49.800
2.5 is off the edge of this chart, I'm just going to have to look at a bigger chart which
00:39:49.800 --> 00:39:52.000
I do not have on the screen right here.
00:39:52.000 --> 00:40:00.900
But my bigger chart shows me that somewhere down here would be 2.5.
00:40:00.900 --> 00:40:07.800
The value is a very small, it is 0.0062.
00:40:07.800 --> 00:40:15.900
That is the probability that a standard normal would be bigger than 2.5.
00:40:15.900 --> 00:40:20.700
That is just a decimal that I threw into my calculator.
00:40:20.700 --> 00:40:37.700
Simplify it out, I got .6853, that turns out to be about 68.53%.
00:40:37.700 --> 00:40:43.800
The way you interpret that answer, by the way is, that is the proportion of students on this exam
00:40:43.800 --> 00:40:51.400
that are going to score between 72 and 96 on this exam.
00:40:51.400 --> 00:40:58.300
The answer to part B, remember from the previous slide, we worked it out to be 1 - the probability
00:40:58.300 --> 00:41:02.600
that Z is bigger than or equal to 2.
00:41:02.600 --> 00:41:22.800
Here is 2.00 on here, which is 1 - 0.0228 which is 0.9772 which is 97.72%.
00:41:22.800 --> 00:41:30.900
That is the passing rate on this exam, that is very fortunate for the students involved, almost all of them passed.
00:41:30.900 --> 00:41:36.000
Remember, the original question there was what percentage of students will get about 60.
00:41:36.000 --> 00:41:43.100
Our answer here is that 97, almost 80% of the students will get above 60.
00:41:43.100 --> 00:41:52.000
To recap what we did on this slide, that was the question that we had from before.
00:41:52.000 --> 00:41:58.500
We converted it on the previous slide, into a couple of values from the Z charts.
00:41:58.500 --> 00:42:03.600
The ½ and ½ put together, that is where I got this 1.
00:42:03.600 --> 00:42:10.100
Z being bigger than 0.5 gave me this 0.3085, that is where that came from.
00:42:10.100 --> 00:42:14.100
Z being bigger than 2.5, that is kind of off the chart here.
00:42:14.100 --> 00:42:17.900
I read that off with a bigger chart which I have, it is not on the screen.
00:42:17.900 --> 00:42:26.100
That is where the 0.0062 came from, simplified that down to 68.53%.
00:42:26.100 --> 00:42:32.500
For part B, we are finding the probability that Y is greater than or equal to 60.
00:42:32.500 --> 00:42:38.300
That turned into a Z value, we did that work on the previous slide, you can check it out.
00:42:38.300 --> 00:42:50.400
The Z value of 2 gives us 0.0228 and that corresponds to 97% of students passing this exam.
00:42:50.400 --> 00:42:56.200
A very happy result for the students there.
00:42:56.200 --> 00:43:01.300
In example 4, we are looking at day time high temperatures in Long Beach California.
00:43:01.300 --> 00:43:03.500
It is a very warm, pleasant city to live in.
00:43:03.500 --> 00:43:09.500
The average high temperature is 75, standard deviation is 9.
00:43:09.500 --> 00:43:13.900
Of course, these are in degrees Fahrenheit, if you are watching from somewhere with degrees Celsius,
00:43:13.900 --> 00:43:17.300
you can convert those if you feel so inclined.
00:43:17.300 --> 00:43:20.900
What percentage of the days has high temperatures above 88?
00:43:20.900 --> 00:43:24.400
How many really hot days we will have here?
00:43:24.400 --> 00:43:29.500
Again, this is a nonstandard normal variable.
00:43:29.500 --> 00:43:38.400
The trick to dealing with nonstandard normal variables is to convert them into standard normal variables.
00:43:38.400 --> 00:43:45.600
Let Z, you define Z to be Y – μ/σ.
00:43:45.600 --> 00:43:49.800
That colon means define to be.
00:43:49.800 --> 00:43:54.900
We are defining our Z right there, to be Y – μ/σ.
00:43:54.900 --> 00:43:57.200
The μ and σ are given to us in the problem.
00:43:57.200 --> 00:44:03.000
The mean is μ = 75, the σ is the standard deviation, it is 9.
00:44:03.000 --> 00:44:08.700
You got to be careful and read these problems, and see whether they are talking about standard deviation or variance.
00:44:08.700 --> 00:44:13.900
Because it is a matter of whether you square or take the square root of the number or not.
00:44:13.900 --> 00:44:20.300
In this case, they are saying standard deviation, that is my σ.
00:44:20.300 --> 00:44:28.400
The point is that, we can convert, we want to find how many days have high temperatures over 88.
00:44:28.400 --> 00:44:31.700
How many really hot days do we have or what percentage?
00:44:31.700 --> 00:44:46.700
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.
00:44:46.700 --> 00:44:52.600
I just plugged in my value of Y into the definition for Z.
00:44:52.600 --> 00:45:06.700
That is the probability that Z is bigger than or equal to, my μ was 75, my σ is 9.
00:45:06.700 --> 00:45:13.900
That is the probability that Z is bigger than or equal to 88 -75 is 13/9.
00:45:13.900 --> 00:45:22.500
And that is the probability that Z is bigger than or equal to 1.44.
00:45:22.500 --> 00:45:30.300
I just threw 13/9 into a calculator there, actually we can do that without a calculator.
00:45:30.300 --> 00:45:38.100
That means that I want to look up a standard normal variable and find the probability that it is bigger than 1.44.
00:45:38.100 --> 00:45:42.800
That is going to be a fairly quick calculation.
00:45:42.800 --> 00:45:47.100
I will just look it up on the chart, there is 1.44.
00:45:47.100 --> 00:45:50.800
I'm looking for the probability Z being greater than that, I want that.
00:45:50.800 --> 00:45:54.100
That is something I can look at directly on a chart.
00:45:54.100 --> 00:45:58.500
I do not need to do any funny area conversions there.
00:45:58.500 --> 00:46:04.500
I set up my chart on the next page and I will just flip over and look that up.
00:46:04.500 --> 00:46:08.800
But let me a recap this, before I'm bury it forever.
00:46:08.800 --> 00:46:13.500
We got a nonstandard normal distribution here.
00:46:13.500 --> 00:46:17.900
The trick to dealing with nonstandard normal distributions is to convert them
00:46:17.900 --> 00:46:22.600
into standard normal distributions, by doing Y – μ/Σ.
00:46:22.600 --> 00:46:26.800
We get the μ and the σ from the problem, you want to be careful to notice
00:46:26.800 --> 00:46:30.400
whether the problem says variance or standard deviation.
00:46:30.400 --> 00:46:33.500
In this case, it says standard deviation, that is my σ.
00:46:33.500 --> 00:46:36.800
I do not need to take a square root.
00:46:36.800 --> 00:46:41.500
I want to know what percentage of days has temperatures above 88.
00:46:41.500 --> 00:46:50.600
Y should be above 88, that means that Z is above, I plugged in 88 for Y, I get 88– μ/σ.
00:46:50.600 --> 00:46:54.200
88 – 75/9 is 13/9.
00:46:54.200 --> 00:46:56.800
13/9 converts to 1.44.
00:46:56.800 --> 00:47:05.700
I’m about to turn the page and look up 1.44 on a standard normal chart.
00:47:05.700 --> 00:47:12.700
What we figured out on the previous page was that, the probability that Y is bigger than or equal to 88
00:47:12.700 --> 00:47:21.100
was the probability that Z, my standard normal variable is bigger than 1.44.
00:47:21.100 --> 00:47:28.100
That should be somewhere on the chart, here is a 1.4 column, the second decimal place is 4.
00:47:28.100 --> 00:47:36.300
Let me read those together, 1.44 gives me 0.0749.
00:47:36.300 --> 00:47:46.300
0.0749, if we convert that into a percentage, that is 7.49%.
00:47:46.300 --> 00:47:57.900
That is the percentage of days in Long Beach that are going to over 88°, really hot days in Long Beach California.
00:47:57.900 --> 00:48:01.500
Just to recap what we did on this page.
00:48:01.500 --> 00:48:07.300
What we are originally asked is what percentage of days are above 88°?
00:48:07.300 --> 00:48:11.700
From the previous page, we converted that into a standard normal variable.
00:48:11.700 --> 00:48:17.100
Y being bigger than 88 corresponds to Z bigger than 1.44.
00:48:17.100 --> 00:48:23.500
Then, I just looked that up on the chart, got my value, and converted it into a percentage.
00:48:23.500 --> 00:48:27.900
If you are lucky in your class, you will be allowed to use a calculator functions.
00:48:27.900 --> 00:48:31.300
You would not need these charts, or some kind of online application.
00:48:31.300 --> 00:48:42.600
If so, go ahead and use those, and do not worry about using these charts because the charts are a little slower.
00:48:42.600 --> 00:48:49.500
In example 5 here, we have, once again, scores on an exam which are normally distributed.
00:48:49.500 --> 00:48:54.200
The mean is 70 and the variance is 16.
00:48:54.200 --> 00:49:01.100
The first thing to notice here is that, you have to be careful, it says the variance is 16 not the standard deviation.
00:49:01.100 --> 00:49:09.400
That means our σ is actually the square root of the variance which would be 4 here, that mean will be 70.
00:49:09.400 --> 00:49:15.100
We want to figure out what percentages of the scores are between 68 and 78?
00:49:15.100 --> 00:49:19.600
Just like the others, the trick to dealing with these nonstandard normal problems
00:49:19.600 --> 00:49:23.400
is to convert them into standard normal problems.
00:49:23.400 --> 00:49:30.100
And then, you can look up the answers for standard normal probabilities on a chart.
00:49:30.100 --> 00:49:39.900
Again, I'm going to set up a standard normal which is always Y - the μ/ the standard deviation.
00:49:39.900 --> 00:49:48.100
In this case, it is Y – 70/ my standard deviation here is 4.
00:49:48.100 --> 00:49:58.100
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.
00:49:58.100 --> 00:50:09.100
And then, I just plug in those values of Y into the formula for Z.
00:50:09.100 --> 00:50:12.100
I want to find Y going from A to B.
00:50:12.100 --> 00:50:21.700
That means A- μ/σ and B – μ/σ.
00:50:21.700 --> 00:50:29.900
That is what I have to do to convert a nonstandard normal variable into a standard normal variable.
00:50:29.900 --> 00:50:44.600
In this case, for part A here, we want to find the probability that Y is between 68 and 78.
00:50:44.600 --> 00:50:48.800
Let us convert that into a probability on Z.
00:50:48.800 --> 00:50:53.300
Z should be between, my A and B are 68 and 78.
00:50:53.300 --> 00:51:14.400
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.
00:51:14.400 --> 00:51:25.800
-2/4 is -.5, 78-70 is 8/4 is 2.
00:51:25.800 --> 00:51:30.900
Let me draw a little graph to show how I plan to calculate that.
00:51:30.900 --> 00:51:35.800
That would be a little bit complicated, I need ½ somewhere.
00:51:35.800 --> 00:51:39.600
I can just fill those ½ coming up.
00:51:39.600 --> 00:51:45.600
You want to go from -0.5 to 2.
00:51:45.600 --> 00:51:50.300
I want this area right here.
00:51:50.300 --> 00:51:55.800
As usual, I think I'm going to split that up into two areas.
00:51:55.800 --> 00:52:12.800
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.
00:52:12.800 --> 00:52:16.100
I’m subtracting off a positive tail because it is symmetric.
00:52:16.100 --> 00:52:19.400
The positive and negative tails are the same area.
00:52:19.400 --> 00:52:30.300
This is ½ - the probability that Z is bigger than or equal to 0.5.
00:52:30.300 --> 00:52:40.600
This right hand area is ½ - the tail, the tail starts at 2 not 2.5.
00:52:40.600 --> 00:52:52.800
½ - the probability that Z is bigger than or equal to 2 + ½ - the probability that Z is bigger than or equal to 2.
00:52:52.800 --> 00:52:54.300
Let me simplify that a little bit.
00:52:54.300 --> 00:53:05.600
Combine the ½ into 1- the probability that Z is bigger than or equal to 0.5 –
00:53:05.600 --> 00:53:11.400
the probability that Z is bigger than or equal to 2.
00:53:11.400 --> 00:53:17.400
I can not move both of those up, as soon as I get a standard normal table for myself.
00:53:17.400 --> 00:53:21.000
I’m going to hang onto those for now and I’m going to look at part B.
00:53:21.000 --> 00:53:27.400
In part B, I want to find the minimum score to be in the top 10% of students.
00:53:27.400 --> 00:53:33.200
Maybe, the top 10% of students will receive honors upon graduation.
00:53:33.200 --> 00:53:36.100
This one is a little different from the previous problem.
00:53:36.100 --> 00:53:38.700
You do not have to think extra hard about this one.
00:53:38.700 --> 00:53:45.400
I want the probability that Z being above some cutoff.
00:53:45.400 --> 00:53:52.700
I do not know what the cutoff is yet, I’m just going to call it z, to be exactly 0.1
00:53:52.700 --> 00:53:56.300
because I want it to be the top 10% of students.
00:53:56.300 --> 00:54:02.700
I want some cutoff and in order to be above that cutoff, you must be in the top 10%.
00:54:02.700 --> 00:54:05.600
That is where I’m getting that 0.1 there.
00:54:05.600 --> 00:54:12.800
I want to figure out what z would have to be, solved for z.
00:54:12.800 --> 00:54:18.600
I have to use that normal chart to figure that out.
00:54:18.600 --> 00:54:25.700
And then, I want to figure out what the cutoff score is for that because that is really going to be the Z value.
00:54:25.700 --> 00:54:32.000
Then, I will get Z is equal to Y – μ/σ.
00:54:32.000 --> 00:54:36.400
Once I figure out what Z is, I can solve for Y.
00:54:36.400 --> 00:54:44.200
I will get Z × σ is equal to Y – μ.
00:54:44.200 --> 00:54:51.400
Y would be equal to Z σ, I will write it as σ Z.
00:54:51.400 --> 00:54:58.300
Σ Z + μ, those solve for Y.
00:54:58.300 --> 00:55:02.300
I can fill the values of σ and μ, I know what those are right now.
00:55:02.300 --> 00:55:11.900
Σ is 4 and my μ was 70, 4Z + 70.
00:55:11.900 --> 00:55:14.000
What I will do is, I will solve for Z.
00:55:14.000 --> 00:55:16.200
I have to do that by looking at the normal chart on the next page.
00:55:16.200 --> 00:55:23.800
Then, I will figure out my cutoff value for Y, after I figure out what Z is.
00:55:23.800 --> 00:55:30.700
That will be the minimum exam score to get you in the honors category.
00:55:30.700 --> 00:55:36.500
Before I go ahead and jump to the Z charts on the next page, let me show you how I calculated everything here.
00:55:36.500 --> 00:55:41.400
We want to find the probability that Y was between 68 and 78.
00:55:41.400 --> 00:55:44.700
Y is a nonstandard normal variable.
00:55:44.700 --> 00:55:50.300
I converted into a standard normal variable here, Y – μ/σ.
00:55:50.300 --> 00:55:53.600
My μ is given by the problem here.
00:55:53.600 --> 00:56:01.600
This σ is variance, the standard deviation is the square root of the variance.
00:56:01.600 --> 00:56:07.000
The standard deviation is √ 16, that is why I used 4 for the standard deviation
00:56:07.000 --> 00:56:14.100
If I have a range of values for Y, here is how I convert it into a range of values for Z.
00:56:14.100 --> 00:56:20.300
I just plug in the A and B for Y into the equation for Z.
00:56:20.300 --> 00:56:24.600
I plugged those in, the A and B are 68 to 78, in this case.
00:56:24.600 --> 00:56:31.300
I got a range for Z, and then, I looked at that on my graph.
00:56:31.300 --> 00:56:35.500
I’m going to split that up and calculate each one of those ranges separately.
00:56:35.500 --> 00:56:45.300
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.
00:56:45.300 --> 00:56:50.000
I'm really calculating ½ - these two tail values.
00:56:50.000 --> 00:56:56.000
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.
00:56:56.000 --> 00:57:03.300
That is how I got this answer that I’m filling the details on the next page.
00:57:03.300 --> 00:57:09.600
For part B, I want to calculate the minimum score to be in the top 10% of students.
00:57:09.600 --> 00:57:20.500
I want to figure out what cutoff on the Z chart will give me a probability of 10% of being above that score.
00:57:20.500 --> 00:57:26.100
I will solve that for Z on the next page.
00:57:26.100 --> 00:57:31.600
Once I find a Z cut off, I will convert that back into a Y.
00:57:31.600 --> 00:57:35.400
I solved this back into Y using μ and σ.
00:57:35.400 --> 00:57:44.500
I figure out a Y cut off that will give me the exam score required to put you in the top 10% of students.
00:57:44.500 --> 00:57:47.200
Let us jump forward to next page.
00:57:47.200 --> 00:57:53.400
Here is my Z chart, and let me remind you what I calculated on the previous page.
00:57:53.400 --> 00:58:02.100
In part A, we wanted the exam scores to be between 68 and 78.
00:58:02.100 --> 00:58:13.000
At some cost of work on the previous page, we converted that into some numbers that we can look up on the Z chart.
00:58:13.000 --> 00:58:25.500
1 - the probability that Z is bigger than 0.5 - the probability that Z was bigger than 2.
00:58:25.500 --> 00:58:34.600
I think both of those are here on the chart, 0.5 is right there, 0.3085.
00:58:34.600 --> 00:58:52.800
.2 is .0228, this is 1 - 0.3085 - 0.0228.
00:58:52.800 --> 00:59:00.800
I reduced those and simplify those on my calculator, I just round it to he nearest percentage.
00:59:00.800 --> 00:59:18.600
I got 0.67 rounded to 67%, that is the percentage of students that will score between 68 and 78 on this exam.
00:59:18.600 --> 00:59:36.700
In the second part of the problem there, we want to find the minimum score above which only 10% of students score.
00:59:36.700 --> 00:59:42.700
The first thing was to solve for cutoff value of Z.
00:59:42.700 --> 00:59:51.800
I want to solve, find z.
00:59:51.800 --> 00:59:59.200
It says that the probability of being bigger than that is 0.10, that is from the 10% given in the problem.
00:59:59.200 --> 01:00:03.000
Let us see, I'm looking for 0.10 in the chart.
01:00:03.000 --> 01:00:06.700
If I look through here, they are getting close to 0.10.
01:00:06.700 --> 01:00:23.600
It is really close to that number there, that is 1.2 and 0.08 is 1.28, that is my Z value.
01:00:23.600 --> 01:00:30.600
Let me write that as, my z is 1.28.
01:00:30.600 --> 01:00:39.900
Exactly 10% of the students or very close to 10% of the students are above a Z score of 1.28.
01:00:39.900 --> 01:00:44.600
But now, I have to convert that back into a y score on the exam.
01:00:44.600 --> 01:01:02.600
My Y, I work this out on the previous page was, 4Z + 70 that is 4 × 1.28 + 70.
01:01:02.600 --> 01:01:11.500
If you calculate that out, it comes out to be very close to 75, just slightly above 75.
01:01:11.500 --> 01:01:22.900
What that means is that, if you score above 75 on this exam then you will be in the top 10% of students.
01:01:22.900 --> 01:01:34.000
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.
01:01:34.000 --> 01:01:35.800
Let me recap here.
01:01:35.800 --> 01:01:47.400
A lot of this work was done on the previous page, converting the values for Y into standard normal values.
01:01:47.400 --> 01:01:51.300
We drew some pictures to figure out that we can calculate the standard normal values,
01:01:51.300 --> 01:01:53.500
in terms of these two cutoffs.
01:01:53.500 --> 01:02:01.600
On this page, we just looked up 0.3085 here and 0.0028 here.
01:02:01.600 --> 01:02:05.100
Those corresponded to 0.5 and 2.0.
01:02:05.100 --> 01:02:09.800
And then, I just did the arithmetic and simplified that down to an approximate
01:02:09.800 --> 01:02:14.200
67% of the students scoring between 68 and 78.
01:02:14.200 --> 01:02:21.600
In part B, we want to find what score you had to get to be in the top 10%.
01:02:21.600 --> 01:02:25.100
This is really sort of a reverse engineering problem.
01:02:25.100 --> 01:02:29.100
We started out with the probability of 0.10.
01:02:29.100 --> 01:02:34.900
I had to find that in the chart and I found it right here, very close to 1.28.
01:02:34.900 --> 01:02:38.000
There is the 1.2, there is the 8.
01:02:38.000 --> 01:02:43.400
That is where I got the Z value of 1.28.
01:02:43.400 --> 01:02:54.700
Again, I work this out, I solve that backwards with a σ and a μ from the previous page.
01:02:54.700 --> 01:03:06.400
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.
01:03:06.400 --> 01:03:14.400
That means you have to score 75 or better on this exam, in order to land within the top 10% of students.
01:03:14.400 --> 01:03:20.000
If you are an honor student, you want to make sure you get those honors by being in the top 10%.
01:03:20.000 --> 01:03:26.400
You better score a 75 or better on this exam.
01:03:26.400 --> 01:03:35.100
That is the last example, that wraps up this lecture on the normal distribution, also known as the Gaussian distribution.
01:03:35.100 --> 01:03:39.600
This is part of the probability lecture series here on www.educator.com.
01:03:39.600 --> 01:03:47.800
The next lecture is on the gamma distribution which also includes the exponential distribution, and the Chi square distribution.
01:03:47.800 --> 01:03:50.900
If you are interested in many of those, I hope you will stick around.
01:03:50.900 --> 01:03:54.000
My name is Will Murray, thank you very much for watching, bye.