00:01
All right, so here we have some exercises involving the central limit theorem and a z distribution for the sample, for a sample mean.
00:12
All right, so we have all these questions are kind of the same in the sense that you have.
00:16
The mean gpa among students is 3 .25.
00:19
So that's our mu, 3 .25, and a standard deviation of 1 .75.
00:27
And we're going to know for 17, for a question 17.
00:30
What's the probability that a random sample of 300 students will have a mean gpa greater than 3 .30? so let's just do a picture of what we're looking for.
00:43
So here's our mean.
00:48
Let's see.
00:49
We'll have a mean gpa greater than 3 .3.
00:52
So if here's our mean, which is 3 .25, that means our 3 .3 is going to be here.
00:59
And we want to know this probability here, this area to the right of our.
01:05
If we do that is we look at a z or z distribution for the sample mean, which is given in the following way.
01:13
Z x bar equals x bar minus the mean of the sampling distribution.
01:18
All over, you might see it in the text as this, excuse me, sigma with a little subscript x bar, which is this thing is called the standard error.
01:29
And it's not this number.
01:30
It's not 1 .75.
01:32
But it is, it's sigma over root n.
01:38
And then i said the central limit theorem because we're not told anything about the distribution, but because we have a sample size with n greater than 30, that means we can assume our distribution is normal.
01:50
And so then we can use this z distribution.
01:56
And from here, oh, and just to be clear, the mean here of 3 .25, that is the same as the mean of the sample of distribution.
02:04
So these are the same.
02:05
So this mean is 3 .25.
02:09
And let's go ahead and do this.
02:11
So for 17, what we do is we take 3 .3, 3 .3, and we subtract 3 .25 divided by 1 .75, divided by route 300.
02:32
And we get the following number.
02:39
Point 4.
02:43
Sorry.
02:46
Point 4949.
03:02
Using just some standard rounding conventions because the number after the fifth place is the seven so we kick it up to it.
03:11
And then this is kind of tricky because you might, depending how you get your values and make sure you're using, this looks like a some sort of a computer program.
03:26
So make sure you check with how you're supposed to read your z scores because a lot of times z table only give you to the hundreds place.
03:37
But i use a spreadsheet to get this, and i can get to quite high precision like this.
03:43
So just be aware of that.
03:45
So what we're looking for is the probability that z is greater than 0 .4949.
03:57
But when we look this up in a z distribution, what we get is this area here, which ends up being 0 .69, 0 .69.
04:11
But we don't want that part.
04:14
We don't want this part...