- 00:01
Hello.Welcome to this lesson of Mastering Statistics.Here, we're going to continue talkingabout using the central limit theorem and population means.And we're going to work with this problem here, whichis also dealing with IQ, something that all of uscan relate to.So, again, the IQ of the populationis-- the mean is 100, the standard deviationof the population IQ is 15.

- 00:23
Here we, again, sample 50 people.And the question is, what is the probabilitythat the average IQ of those 50 people that we sampledis greater than 105?So it's exactly the same problem that we did in the last lessonthere.Except now, instead of finding the probabilitythat a sample of 50 is less than 95, now we want to see,what is the probability that the average IQ of those 50 people

- 00:45
is greater than 105?So it's very, very similar.In fact, that's why I chose it here.What we know, though, is that the mean of the samplingdistribution-- if we choose samples of 50 people--is going to be equal to the population mean, which is 100.So we can write that down.We also know that the standard deviationof the sampling distribution is going to basically end up

- 01:06
being the same thing.Remember, it is the standard deviationof the population divided by the square root of the samplesize, which is 15 over the square root, in this case,of 50.And just as we got last time, 2.122,which is exactly what we got last time because noneof those numbers have changed.But let's just take one quick second

- 01:27
to draw a picture of what we're looking at nowso you have a mental image of howit's different from the last problem.This will be the sampling distribution of sample means.We know that it's going to look normalbecause the sample size, n, is 50, which is greater than 30.So when we take 50 people, 50 people, 50 people, 50 people,

- 01:47
and calculate their average IQ and collect all that data,we expect to get something normal shaped.And we expect the sample mean of the sample distributionto be 100.And we expect the standard deviation of this distributionto be 2.121.Now, the question here, though, it'sasking us, what is the probabilitythat the sample of these 50 people

- 02:08
will have an average IQ of greater than 105?So this is 100.So let's just go up a little bit.We'll call this 105.And that's kind of hard to read there.And we're asking, what is the probabilitythat a sample of 50 people will be greater than 105?So another way of phrasing that is saying,look, we create a sampling distribution of samplemeans where there's 50 people in each sample.

- 02:29
The sample size is 50.What is the probability that a samplemean is greater than 105?That's basically what we're asking.So this is a distribution of sample means.We choose 105.We want to find the area under the curve to the right.All right, so we have to use a z-score.And the formula that we've been using hereis x bar minus the mean of this distribution divided

- 02:51
by the standard deviation of this distribution.So what goes here is the sample mean that we care about,which is 105, minus the actual meanof this normal distribution, 100.And then we have 2.121.So when we get this guy, we're goingto get-- this is a positive number now.105 minus 100 will give you a positive number.2.36.

- 03:13
So really, in order to solve the problem,we want the area to the right.So really what we're looking at is the probabilitythat z is greater than 2.36.That's what we want.But our z-tables are not set up to calculate the probabilityto the right.And we ran into this in previous lessonsa long time ago when we started using the z-chart table.They're always going to give you the probability to the left.

- 03:35
So any time you're asked to find the probabilityto the right of a number, you haveto rephrase it as the probabilitythat z will be less than negative, in this case, 2.36.So you flip the direction of the side.And you stick a negative on this.And this comes about from the symmetryof the normal distribution because the areato the right of this is the same as the area to the left of zero

- 04:03
going the other direction.We talked a lot about that when we first learnedabout normal distributions.When you're trying to find the probabilitygreater than the z-score, you justsimply flip the sign around, stick a negative on this guy.So really what we want to do is look up negative 2.36in our table.And read directly off the chart.You get 0.0091.

- 04:24
Now, doesn't this look familiar?This is the probability that, if we take 50 people and averagetheir IQ, that the answer that we getis going to be greater than 105.OK, and the answer is pretty small probability.This number that we get here is exactly the sameas what we got in the last problem.In the last problem, we were trying

- 04:45
to find the probability that the average IQ of 50 peopleis less than 95.Why are these two numbers giving us the same answer?And that comes about because of the symmetryof what we have here.I'll just quickly redraw.Down here, I'll redraw this.And I'll redraw this.You have to use your imagination.I'm doing this kind of quickly here.

- 05:05
But what we found in this problem-- this is 105.We found this area right here.In the last problem, we found-- this is 95.We found the probability that the IQ is less than 95,of those 50 people.And what we're saying here is that the answerwe got for this shaded area was 0.0091.In the last problem, the answer we got for this shaded area

- 05:27
was also 0.0091.Why are they exactly the same?Well, that's because the mean of this distributionis 100, all right.We've established that from before.95 is five units less.105 is five units greater.So because the normal distributionis exactly symmetric, and because thisis five units up and five units below,

- 05:48
the area here is going to always match the area hereif you choose values that are symmetric about the meanlike that.And that's why you get the same answer.So it's kind of carefully chosen.I wanted to do the calculation with youto show you how to get the answer.But the answer also makes sense with whatwe have calculated previously.So the answer we get, 0.0091.

- 06:10
That's the probability.Now, the next question is kind of a follow on to this.If 50 people are sampled, what is the probabilitythat their mean IQ will differ from the population meanby more than five?What is the probability that their mean IQwill differ from the population mean by more than five?

- 06:32
As with any type of word problem,95% of what you need to really work onis understanding what the problem's telling you.So when you read a problem, especially in statistics,you need to make sure you understand what it's saying.And draw a picture because, a lot of times, drawing a picturewill help you understand it.Even if you thought you understood it the first time,

- 06:52
drawing a picture will help it.So here what we're saying is we knowthat we've sampled 50 people.We know that, if we do that, we'regoing to get a sampling distribution of sample means.And we know that it's going to look normal.So let's draw that.Now, what it's asking us is, if 50 people are sampled,what's the probability that your mean willdiffer from the population mean by more than five?Now, the population mean is 100.

- 07:14
We've established that many times over,100 right here in the center.The sampling distribution of the sample meanshas a mean, which is the same as the population mean, whichis 100.So we write that down.It's saying, what is the probabilitytheir mean will differ by more than five?So here, this is 105.And over here, this is 95.

- 07:37
So what we're really asking is, whatis this shaded region here plus this shaded region here?Would you agree that this shaded region-- both of themtogether-- represents the probabilitythat a random sample of 50 people, if you averagetheir IQ, that their mean will differ-- which means differ,

- 07:58
I like this, to the left and to the right-- from the populationmean by more than five?So this is more than five this way.And this is more than five this way.So we need to get them together.Now, we already know-- because we've alreadydone the calculation, really-- that this area is 0.0091.And we've already done the previous problem before.We know this area is the 0.0091.

- 08:20
So, really, the probability that we care aboutis 0.0091 multiplied by 2.So the probability is 0.0182.And that's the answer there.So, like I said, a lot of times reading the problem,drawing the problem is going to help you really and trulyunderstand it.And that is what we're doing here today.So I hope you're getting a little more

- 08:41
familiar with the central limit theoremand applying it to population meansand doing these types of problems.None of the math is difficult. There'snot any fancy, higher-level math here.Understanding the problem is paramount, though.So make sure you understand these problems.Then follow me on to the next lesson.We're not done yet.We're still going to do more problemsto get our practice with the central limit theorem.

- 09:01
And then we'll expand its use to other types of problemsas we move forward.

### Video Info

**Series Name:** Mastering Statistics, Vol 3

**Episode:** 5

**Publisher:** Math Tutor DVD LLC.

**Publication Year:** 2014

**Video Type:**Tutorial

**Methods:** Normal distribution

**Keywords:** mathematical computing; mathematics

### Segment Info

**Segment Num.:** 1

**Persons Discussed:**

**Events Discussed:**

**Keywords:**

## Abstract

Jason Gibson provides example problems on applying the central limit theorem to the population mean.