## Normal Distribution: PDF vs. CDF

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## Mathematics: General Statistics

## Transcription: Normal Distribution: PDF vs. CDF

*Hi and welcome to www.educator.com.*0000

*Today we are going to be talking about normal distributions again but this time breaking it down into the PDF *0002

*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 before *0018

*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 people *0102

*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 out *0226

*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 distribution *0336

*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=x ^{2} 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 continuum *0495

*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}/2x^{2}.*0674

*That is the heart of this and even this ½ is like a constant, I just remember it as e^-x ^{2}.*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 x ^{2} or –x^{2}/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 www.wolframalpha.com 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 www.google.com, www.wolframalpha.com 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 x ^{2}.*0778

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

*What www.wolframalpha.com 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 x ^{2} 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 mean *0966

*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 x ^{2} 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 – 0 ^{2}.*1020

*That is what we could just convince it is x ^{2}/ 2Ʃ^{2}.*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 Ʃ ^{2}.*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 www.wolframalpha.com 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 × Ʃ^{2}.*1135

*I’m just going to make that 100 ^{2}, 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 × Ʃ ^{2}.*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 www.wolframalpha.com is that just in case I missed a parentheses *1204

*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 score *1777

*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 10.13.37.*2460

*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 here *3216

*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 www.educator.com.*3342

0 answers

Post by Mohammed Alam on August 6 at 07:18:45 PM

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 :(