Probability 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 10326

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 in0490

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 table0784

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 out0801

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 probability0824

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 necessarily0895

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 you0982

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 probability1007

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 see1020

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 and1037

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 -21480

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 mean1575

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 practice1647

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 exam2438

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 them2773

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