Sign In | Subscribe

Enter your Sign on user name and password.

Forgot password?
  • Follow us on:
Start learning today, and be successful in your academic & professional career. Start Today!
Loading video...
This is a quick preview of the lesson. For full access, please Log In or Sign up.
For more information, please see full course syllabus of Statistics
  • Discussion

  • Download Lecture Slides

  • Table of Contents

  • Transcription

  • Related Books

Lecture Comments (4)

0 answers

Post by Mohammed Alam on August 6, 2014

Useful lecture series. Good instructor.

0 answers

Post by Manoj Joseph on June 17, 2013

I am finding difficult to make sense this session.

0 answers

Post by Robin Dorsey on October 20, 2012

This example assumes that quarterly results with have a normal distribution...why is this a reasonable assumption?

0 answers

Post by Kamal Almarzooq on January 23, 2012

too much depend on excel makes your teachings less useful to me :(

Normal Distribution: PDF vs. CDF

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Roadmap 0:15
    • Roadmap
  • Frequency vs. Cumulative Frequency 0:56
    • Frequency vs. Cumulative Frequency
  • Frequency vs. Cumulative Frequency 4:32
    • Frequency vs. Cumulative Frequency Cont.
  • Calculus in Brief 6:21
    • Derivative-Integral Continuum
  • PDF 10:08
    • PDF for Standard Normal Distribution
    • PDF for Normal Distribution
  • Integral of PDF = CDF 21:27
    • Integral of PDF = CDF
  • Example 1: Cumulative Frequency Graph 23:31
  • Example 2: Mean, Standard Deviation, and Probability 24:43
  • Example 3: Mean and Standard Deviation 35:50
  • Example 4: Age of Cars 49:32

Transcription: Normal Distribution: PDF vs. CDF

Hi and welcome to

Today we are going to be talking about normal distributions again but this time breaking it down into the PDF0002

or probability density function and CDF or the cumulative distribution function.0008

Here is what we have for today.0014

We are going to be talking about frequency charts which we have been doing before0018

and then contrast it to this new idea but still fairly elementary, cumulative frequency of charts.0022

We are going to do a very brief review of calculus.0031

It is going to be a very elementary review, no actual calculations just conceptual.0034

Then we are going to talk more deeply about the probability density function and the cumulative distribution function.0042

You have to reprogram yourself to see PDF and not think of it as a portable document or format.0048

Let us talk about frequency versus cumulative frequency.0056

So far what we have been doing is we have some sort of variables such as score on SAT verbal.0061

We have all these values that value can potentially hold.0067

We have been talking about what percentage of our sample or population have achieved that score.0072

1% score is 800, 3 score is 750.0081

That is what we have been talking about so far.0086

Here when I write percentage I am talking about relative frequency but it is largely the same thing as frequency.0089

Not a big deal.0100

When we talk about cumulative percentile, what we are really talking about is not just the people0102

who has achieved that value but an accumulation of everybody who has come before it.0112

Let us start off at the bottom.0118

Here only 1% of the population has achieved 250 points.0120

It think that is one of the minimum or something.0126

But 3% have achieved 300 or below.0129

If you are in the third percentile, you have out performed 3% of every else that has taken the test.0136

Not something to write home about yet.0144

This 8% actually accounts for this 3% as well plus a little bit extra.0147

The 16% encapsulates everybody who has come before it.0163

Cumulative percentile is helpful is you want to know your ranking in a performance.0169

For instance you want to know what percentile of the population you are in.0175

You just do not want to know what percent of everybody who is around you has also achieved that score.0179

You want to know how many people you have out performed.0185

Cumulative percentile continuously adding all the people that have come before you.0188

It gives you that ranking.0195

If you are in the 95 percentile, you know out performed or equally 95% of that sample.0198

That is cumulative frequency is really helpful to us.0210

One of the things that you want to notice is that cumulative frequency, when you just look at the number right away,0215

you do not know how common that score is.0222

When you look at 98% you do not know how common 750 as a score is but you could easily find that out0226

just by looking at the difference between 98 and whatever cumulative frequency came before it.0235

That difference is 3, so 3% of people have been in that bracket.0240

Another thing about cumulative frequency I want you to notice is that it is a monotonic increase.0247

It means that there is no going up and then going back down.0254

There are no changes in direction.0260

It is continuously going up, up, and up.0262

That makes sense because you have to add up everybody who has come before you.0265

That is cumulative frequency.0269

When you look at it on a visualization, you could see what I mean by monotonic increase.0272

Here we have an example of monotonic increase.0280

This curve goes up, and up, and up because it is adding up everybody who has come before you.0286

You have to be at least the score before you or higher.0293

Every score is improving on the previous score.0298

Whereas in frequency we can have non monotonic curves because here you could go up, down, we could have the uniform distribution.0304

We could have all kinds of things but in cumulative frequency distribution you cannot have a uniform distribution.0315

If everybody has only see the bottom score then you would have a uniform distribution because you will be adding 0 everytime.0321

Otherwise the most frequent shape that you will see is a monotonic increase that looks like this.0329

Here we see just like this normal looking distribution and what ends up happening when you have this normal-ish distribution0336

you have the s shaped curve when you transform it into the cumulative frequency.0345

This part in the middle, that part corresponds from about 400 to 600.0352

That part corresponds to the biggest jumps and as you see there are big jumps here too but the jumps are just going in one direction.0362

That is how it looks.0377

Let us put that on hold for a second and let us talk about calculus in brief.0382

A lot of people when they think about calculus, they think immediately derivatives and intervals like integrating stuff and getting derivative of equations.0397

It is what they think about.0398

Let us start packing and think about it conceptually.0399

Calculus in some way you could think of an equation as being on a continuum.0403

Let us just say that there are some equation that we are thinking of Y=x2 or something like that.0409

We all know what that function looks like.0418

It looks something like this a parabola and when we think about this we are saying this is just plotting exactly what y is given from x.0421

That is all that graph is.0434

When x is 0 y is 0.0435

When x is 1 y is 1.0437

When x is 2 y is 4.0439

We are just plotting those precise points.0442

When you go and take the integral, I’m going to put the integral on this side versus you go and take the derivative,0447

you are describing 2 different aspects of the same graph.0459

When you talk about the integral, what you are doing is you are no longer plotting these particular point but now you are plotting these areas.0467

When you think about integral, think about area.0480

Whenever you have some curve or shape or line, when you take the integral you are pointing towards the area.0485

You could get these areas of this weird bizarre shapes and at the same if you go on the other side of the continuum0495

and you go towards the derivative which you will end up getting is not the area but instead you are just getting the slope.0504

Here instead of being interested in the points themselves, now you are interested in these little slopes.0516

All these little slopes, that every single, tiny point.0526

You are really plotting those slopes.0531

You are interested in these changes.0536

Here I want you to think slope and obviously for any equation so even when you get a graph of slopes, you could get the slopes of slopes.0540

Even when you have the slopes of slopes, you could get a slope of slopes of slopes.0552

You could go more and more towards that derivative side or on the other hand if you get a graph of whole bunch of areas you could get the areas of that curve.0556

You could go more and more towards the integral side as well.0566

This is going to be important to us because we are going to be interested particularly in a curve that looks like this, the normal curve PDF.0575

What we want to do is get the cumulative areas of that one.0585

Which direction should we go?0592

We should probably go towards the integral direction because what we want is the cumulative areas of this function.0595

That is going to be an important concept.0605

Now let us talk about the PDF or what we call the probability density function.0610

We have talk about how the standard normal distribution is a little bit different than just the normal distribution.0617

The mean is 0 and the stdev is always one because of that it is a special case that is very helpful to us.0622

There is a special sign we use just for the PDF of the standard normal distribution.0638

That is what we call little yi and it looks like this.0644

When it is written out sometimes it looks like a lower case y.0648

That is how people sometimes write it.0654

That is what it looks like.0657

That is the symbol we use to denote that we are going to be talking about standard normal distribution.0659

The function actually looks something like this.0667

X just represents some value on your variable of interest and there are different ways you could write this but the heart of this is e-1/2x2.0674

That is the heart of this and even this ½ is like a constant, I just remember it as e^-x2.0683

That is the heart of the shape, all divided by 2pi.0694

Obviously this can be written in slightly different ways you might also see it as 1/√2pi multiplied by the exponential function to the power -1/2 x2 or –x2/2.0715

There are couple of different ways you could write that.0735

There function might seem a little bit crazy to us but let us break it down.0738

You might want to use or if you have a graphing calculator feel free to use that.0743

That will work largely in the same way.0750

If you go to, is like a combination of fancy graphing calculators / Wikipedia for math and science stuff.0754

It is helpful.0767

Here we could write any function we want.0771

Let us start off with just y= e^ -.5 x2.0778

Let us start with that part first and let us see what we got.0802

What will do is that it will actually draw the equation for us.0807

Notice that we have something that looks like a normal distribution but one of the issues is we just want to divide it by the √2×pi.0811

That will just change the shape of it very slightly.0834

Notice that largely it is the same basic fundamental shape.0840

You divide by that constant to give you a couple of properties of a normal distribution that are going to be important to us later.0844

The heart of this function is that exponential function and it is in particular exponential function to the x2 power.0851

That is what gives us that nice curve.0866

That is what we think of little yi, that is this equation.0871

The PDF for a normal distribution that we know has any kind of mean and any kind of standard deviation.0877

One thing that I forgot to point out is that when you look at this, one thing that you will notice is that then mean or point of symmetry is 0 and 1 stdev out.0888

If you go 1 stdev out that looks about like 68% of that curve.0913

It seems like more than half.0923

This depicts the mean being 0 and stdev being 1.0926

If we want a regular normal distribution that did not have a mean of 0, it did not have stdev of 1, what would the formula for that be?0934

What would the function for that be?0948

The general equation for the PDF, we do not have a special symbol, we just use the regular symbol that we use for all function f(x).0951

Here it is still this function at the heart of it but all we are doing is we are going to be adding in mean0966

and stdev as variables so that you could put in whatever mean and stdev that you want.0975

Let us start with the heart part.0981

It is e^ -, instead of ½ we need to change that x2 and it is going to be x – mu because we are going to put in that mu.0985

If mu is 50 we want that point of symmetry to be over 50.1005

If mu is -5 we want that point of symmetry to be over -5.1010

We want to square that.1015

We are squaring that distance.1018

Here we have done that actually except that is x – 02.1020

That is what we could just convince it is x2/ 2Ʃ2.1030

That is where that ½ comes from and when Ʃ is 1 it is just -1, right?1038

That is why we do not see that crazy part in here.1046

There is just one more thing, 2 pi Ʃ2.1051

Another way you could think about it is having 1/Ʃ × phi, instead of just putting an x we are putting in x – mu / Ʃ.1059

If you out this function in here and you substitute x with all of these crap and you multiply by 1/ Ʃ then you would get this equation.1080

This is the simplified version.1093

Now let us put this in and let us see what kind of function we have.1096

We could put in a mu(550) and Ʃ(100).1102

Let us see what that normal distribution looks like.1114

I will put this up here.1120

We want to substitute in 550 right here and 100 right here.1127

Let us put in e^ - ( x – mu(550))2 ÷ 2 × Ʃ2.1135

I’m just going to make that 1002, that is 10,000.1160

I’m just going to make sure that it is all in this parentheses here.1166

All divided by √2×pi × Ʃ2.1179

I’m just going to put my 10,000 again.1191

What we should see from this is the mean being at 550 and the standard deviation being at 100.1195

Let us see and the nice thing about is that just in case I missed a parentheses1204

it will rewrite the input for you in a more standard form instead of this linear way.1220

So that you could check and see that you are missing a parentheses or something.1229

Is this is where 550 is and that looks like the center of that curve and not only that from 550 if you go out to 450 or 650,1234

that does looks like about 68% of that curve.1249

What we see here is that this function, if you substitute in any mean and any stdev it will effectively draw or represent this normal distribution for you.1253

That is the PDF, but what this gives you is at every point what is the probability for that particular value of x.1269

Let us move on to cumulative distribution function.1281

We want the cumulative distribution function.1287

We do not just want that curve.1292

Instead what we want is a cumulative adding up of all the areas that came before.1298

What we are looking for is that curve and at any point I can tell you the percentage we are at.1306

Here, it encapsulates all the space that came before it.1314

As we talked about before, because I want that cumulative area, all we have to do now is take that integral of phi.1323

We represent the cumulative density function as upper case phi rather than lower case phi and we put in x.1331

All we do is we take the integral and we take the integral from –infinity up to x, whatever x is and that will give you the area so far of phi.1348

Obviously you could actually get the integral but I’m just to leave it as it is because I just want you to know what the function actually means.1367

The meaning is just the integral of little phi which we talked about.1380

What that gives you is this idea of the area up to this point of x, whatever x is.1387

Because we are talking about for a standard normal distribution that is why I’m using that phi or else I would use f(x) for the normal distribution function.1395

That is PDF.1407

That is a little bit easier.1409

Let us go into some examples.1412

Here is example 1 and it is just talking about frequency graphs.1414

It is not actually talking about CDF or the function.1419

Here it says that estimate a square that falls up to 48% percentile.1424

Percentile is a word that we say just for cumulative frequency or cumulative relative frequency.1430

One thing that makes this chart easy to use is that we could just go to 48, I will use a different color and I will go all the way across and go down.1438

We will get a rough approximation of that score.1454

That score is about a little bit less than 500, let us say 480.1459

That is the score that certifies the 48% percentile.1465

If you wanted it a little bit more than 40, you could just round it to 500.1470

If you want it less than 48% percentile, you could round down to 450.1474

Here is example 2.1485

In problems with a normal distribution, the mean, stdev, x and the probability of x, these things are involved.1486

Like a puzzle, if you have 3 you could figure out 4, right?1499

Here we have couple of things that are missing but it gives you some of the other pieces and we have to figure out the other pieces that are missing.1505

I’m just going to take a look at my first line.1514

It says the mean is 3, stdev is 1, I have an x value, what is the probability of that x?1519

I’m just going to pull up a regular Excel sheet and I have just labeled it with mean, stdev x, and probability of that x.1528

I’m also going to write down z because often we need to find z in order to use the tables at the back of the book.1547

I’m going to write in z.1556

The mean here is 3, stdev is 1, x is 2, if we taught about that in a normal distribution picture.1561

The mean will be 3, stdev 1, this will be 2.1586

Since x is that 2, it is asking for this.1587

It should be about 16%, just using that empirical rule.1593

Let us find out exactly how much that is.1603

If you wanted to use the special Excel function, you could actually just use normdist because you have everything you need.1607

You have your x, mean, stev, and if you want the cumulative probability you would just write true which is what we want.1618

We will get about 16% or .159.1634

Another way you could do it is by finding the z score and then looking in up at the back of your book.1640

You could use standardized or you could find the distance between the x and the mean and divide that by the stdev to get how many stdev away.1648

It is 1 stdev away on the negative side.1663

If I did not want to use this function I could just use normdist and that is where I will put in my z score and I will get the same thing.1667

Those are 2 different ways that you could do it.1683

You could also look at that z score at the back of your book.1685

Let us do the second line.1690

Here we have the mean but we do not have the stdev but we do have x which is .1 and 1x < .1 it is at .18.1691

Let us sketch this out to help ourselves and I will draw it in a different color.1713

Here the mean is 10 but I do not know what to write here.1723

What I do know is around 18% which is a little more than 16%, around 18% that x is .1.1730

My question is what is my stdev?1744

What is this jump such that this 18% that is .1?1751

We have all the pieces that we need, one thing that might be helpful to do is find the z score because the z score formula has the stdev.1756

We have all the other pieces like the mean and x.1768

Once we know the z score then we could easily find out the stdev.1771

I’m just going to use my norm inverse which would get if I put in the probability it will give me the z score1777

or you could also look up in the table at the back of your book and look for 18% and find the z score there.1787

Here I find my z score is -.91.1798

It is a little bit closer to the mean than -1 but it is almost -1.1804

Once I have that then I could easily find my stdev function just by using my z score formula because the z score formula is just x – mu / stdev or /Ʃ.1813

What I want to solve is for Ʃ and here I could just match my Ʃ and both sides multiply and divide z by both sides and I could get Ʃ = x – mu / z score.1838

Then I could easily do in Excel or in your calculator or in your head.1857

I will just take x – mean / z score and I wil get 10.81.1866

That makes sense because if I went out about 10.81 that will be -.81 right here at the first stdev.1881

This is much smaller than that.1893

Here that answer makes sense.1898

Let us go to the third problem.1902

Now I’m missing my mean but I have my stdev, I have m y x which is -.6 and I know that probability where x < -.6 is 35%.1905

I will put in that.1926

Let us sketch that out just so we can check whether the answer that we get is reasonable.1929

Now we have an idea what this middle value is but we do know that each jump is about 3 away.1937

The other we cannot tell where to write that .6 we could tell by the percentage it cannot be up here.1946

It must be somewhere here such that this is about 35%.1958

It is not quite half of that half but it is a little bit more than half of this half.1975

The question is what is this mean?1985

We know at that point it is -.6.1988

Whatever my mean is, it got to be bigger than -.6 just because we know that this is not quite half yet and if we have the middle that is the mean.1994

Once again it is half way to find the z score because the z score formula has the mean in it and so it is easy to find the mean once we have the z score.2008

Once again I’m going to use norms in and put in my probability and I will get the z score of -.38.2020

That makes sense.2036

That is in between 0 and -1.2037

Let us use that in able to find our mean.2043

If you want you could just derive it from this again so instead of going from z, x but now we want to isolate mu.2046

It would be z Ʃ = x – mu and I’m going to move the mu over to this side and move this over this side.2060

That is what I need to do in order to get my mean, mu.2079

I will take my x and subtract the z score × stdev and I will get mean of .56.2088

That makes sense because that is bigger than .6 and since that there is a distance of 3 that would makes sense for .6 in between .55 and -2.55.2106

This makes sense to us so once you write the mean, the answer, you are good to go.2137

Now let us move on to example 3.2146

It is another puzzle like problem but once you know the mean and the stdev you could find out Q1 and Q3.2153

If you want Q1 and Q3 that could hel you figure out the mean and stdev.2161

Here we are missing Q1 and Q3.2167

Here we are missing the mean and stdev.2171

And here I am missing a little bit of both.2174

Let us get started.2177

Here let us start just by drawing what we are talking about here.2180

Here we have the mean and the stdev which is 5, 5 away on this side and 5 away on this side.2186

We know that at about this little score right here we know that is 34% of the curve.2197

Q1 wants to split it out into quartiles not 16 and 34, that is not even enough.2207

We know that Q1 has to be somewhere in between 1 stdev away and 0 stdev away.2216

We know that it means to be somewhere in there.2225

Not quite stdev away because we want this area which makes this 25%, makes that 25%, and same with Q3 and up here.2229

That is what we are looking for here.2246

I will show you 2 ways of doing it just with Excel.2250

Excel makes it a lot easier for us to do these things.2256

Here I have mean, stdev, Q1, and Q3.2260

We have the mean of 10 and stdev of 5.2271

One thing that is helpful is just to know what the z score is in Q1 and that is never going to change because z score is just how many stdev away.2276

You could think about z scores as just being a reflection of the standard normal distribution and that never changes.2287

The z at Q1, what is that?2294

That is easy to find by using the norms inv function and there you would just put in the cumulative probability you want.2298

That would be this area right here which is an easy 25%.2308

I know my z score there, it is not quite -1 it is -.67.2314

That makes sense.2321

Once we know that, then we have all the things we need in order to find the raw score at Q1 or what we call x.2323

We could put in the mean + z score × stdev and because Excel always holds order of operations, it will do the multiplication before it does the addition.2331

That is going to be 6.63 and does that make sense?2351

Yes it does.2357

It is in between 5 and 10 and it is pretty close to 5 but not quite all the way to 5.2359

That makes sense that would be 6.63 and we could do the same thing in order to find Q3.2366

Let us find the z at Q3.2376

If you wanted to do this without Excel you could also easily do that too because you know that the z at Q3 if you cover 75% of that curve.2380

This is also .25, if you add it all up that is 75% of that curve.2394

You could just look that up in the table at the back of your book.2402

Look for .75 and then look for the z score that correspond at that point or you could find that in Excel by using norms.2406

It is important to have that s because if you are just looking for the z score and put in the probability of 75%.2419

.75 gives us .67 as the z score and that makes sense that these z scores for Q1 and Q3 precisely mirror each other.2427

They are just the negative and positive versions of each other.2438

For Q3, we could just use the mean and add how many stdev away you want to go.2442

That is z × stdev and that gives us 13.37.2453

If we look on this side it is not quite 15 but it is closer to 15 than it is to

That makes sense.2471

I will write that in here 13.37 and this is something like 6.63.2472

That is finding Q1 and Q3.2485

There is yet another way that you can do it in Excel and this is going to be a super short cut.2489

You could use the norm inv function because you have the probability .25.2496

You have the mean and the stdev.2505

That will give you the z score.2510

You could also use that for finding Q3 by using norm inv where you put in the probability and I will spit out the raw score.2513

The probability is 7.75, mean is 10, stdev is 5, and once again we get the same thing.2522

For a lot less work you do not have to go through the z score method.2535

The z score method is helpful just because you could also use the back of your book and maybe on your test who have Excel.2538

That is a good and helpful thing to know.2547

Now let us move on to the second problem.2551

The second row the problem has changed a little bit.2555

We have the same curve but we know that we do not know the mean, what that jump is, but we do know this.2560

We know here the score is 120 and we know here the score is 180.2575

I’m just going to show you a quick short cut here but it is very reasonable shortcut.2585

We know that the mean has to be in the middle of these 2 numbers.2589

Those 2 numbers are are mirrors of each other.2594

They are exactly 25% away on this side and 25% away on this side and the normal distribution is perfectly symmetrical.2597

We k now that the mean has to be the point in the middle.2603

There is only one point that is precisely in the middle of those 2 numbers and we could easily find that by taking the average of 120 and 80.2608

Between 120 and 80.2619

I could just say take the average of these 2 numbers or you could alternatively add 120 to 180 and divide it by 2 and we would get 150.2622

150 looks about right.2638

That looks like where it should fall.2642

The question now is what is the stdev?2647

One way you could easily do this is we know the z score and we could use it to figure out the stdev.2653

I do not have to find the z score again.2667

I’m just going to use these.2669

Now let us think about the formula for stdev.2672

If you remember from the previous problem, it is just going to be whatever x in.2675

I’m just going to use Q1 as my x – mean / z score.2684

I will get 44.48.2691

Let us see if that makes sense to us.2695

If we are at 150, if we go out 44.5 then that should give us about 105 or 106.2700

If we go out that far that makes sense because 120 falls in between that and 150 but it is a little bit closer to the 105 than it is to the 150.2726

If we go out on the other side it will be 194.5.2741

Once again it makes sense because 180 is pretty close to it but not all the way up there.2748

Last problem in this set.2754

Here we do not know the mean but we do know the stdev, the jump.2760

This is a jump of 10.2770

We know Q1.2772

Here is Q1 and that is 100.2776

We know that the mean has to be greater than 100.2784

We do not know exactly how much greater but we know it is greater than 100.2790

It cannot be 110 because you are not going 1 stdev out, you are going less than 1 stdev out.2797

Let us see if we could figure out this strategy.2806

We could find the z score very easily but we already know it.2810

Using the z score we could find the mean.2814

Once we know that we could find Q3.2818

I will move this up here.2821

Here we do know the mean but we do know the stdev.2829

We know Q1 and we know the z score at Q1.2833

Using that z score in Q1 I’m just going to go ahead and find my mean.2838

In order to find the mean, if you remember from the previous problem it is just x – stdev × z score.2844

The mean is at 106.75.2863

That makes sense.2873

It is not quite 110 but it is in between there and once we know that mean 106.7 then we could easily use that in order to find Q3.2876

Q3 we could use norm inv and put in probability, mean, and stdev, or you could use the z score in order to find Q3.2909

That makes sense 113 because if the mean is 106 or 107, then going 10 out would be 116.7 and that is too far out.2932

113 is perfect for Q3.2952

That is example 3 and notice that it just takes a little bit of reasoning to get around some of these things.2956

Let us go to example 4.2972

The miniature cars in an old town is 12 years old and the stdev is 8 years, what percentage of cars are more than 4 years old?2975

One thing that helps me is if I draw a little distribution to help me out.2984

The mean is 12, stdev is 8, what percentage of cars are more than 4 years old?2992

There is implicit issue here.3024

Let us say we go another 8 out that would mean we are at -4.3031

Can a car be -4 years old?3034

Let us think about this.3037

It looks like maybe in somebody’s head or in plans.3040

-4 years old is hard to think about.3047

When we think about this distribution we want to cut it off at 0 because cars just are not -1 years old.3052

It starts at 0.3068

If we think about where 0 is, that is right in between there.3070

When we think about it in terms of z score, the z score is 0, this is -1 and this is about -1.5.3076

We are thinking about we do not want to count these cars because they do not exist.3087

Thankfully our question is what percentage of the cars are more than 4 years old.3099

Our question is a trick question to help us think about this issue.3107

We are asking for this but remember percentage is always what is that compared to the whole.3117

The whole is a little bit different here.3125

The whole is not this whole curve because this part does not count.3128

The whole is actually this part.3135

It is asking what is the blue part in proportion to the red part?3144

Tricky question.3150

This takes a little bit of thinking.3152

One thing probably we want to do is figure out the proportion of cars where age is greater than 4 years old and divide that by the proportion of cars where h > 0.3155

That we could easily do by using z scores.3181

We could take the z score or p(z score) > -1 / p or the z score > -1.5.3184

You could do this by looking these probabilities at the back of your book or we will find these probabilities in Excel.3202

Here is p where z > -1 and remember because we want the greater side, we have to do 1 – the functions here3216

because Excel will give us the part on the negative side.3235

Instead of the greater than side it will give us the less than side.3240

I could use normsdist where we out in the z -1 and then it will split out the probability but it will split out the less than probability.3243

Here I want to put in the 1 – normsdist and this should be greater than 50%.3258

It should be 80 or something percent.3266

Let us find the probability where the z is greater than – 1.5.3271

I could use that same function normsdist of -1.5.3281

Once again because I want the greater than part, I want to use my 1-.3289

Once we have that then we could get the proportion what percentage of cars is here over here.3296

That would be this over that.3306

That is 90% of cars.3314

This will be 90% of cars.3318

This seems to be a little bit of a tricky problem.3322

It does not look tricky at first but watch out for things like this whether it is a cut off at 0.3326

You cannot have a negative age in this case.3333

Watch out for these problems.3339

That is it for