00:01
For this problem, we're looking at a population that we're told has a mean mu of 50 and a standard deviation sigma of 10.
00:12
And from this population, we're going to take a random sample of size 36.
00:16
So let's say that n is 36.
00:20
And given all this information, we want to answer a few questions.
00:25
So first, part a, we want to describe the shape of the sampling distribution.
00:33
So let me just say that our random variable for the population is x, then our random variable for the sampling distribution of size 36 will be x -bar.
00:44
And we want to describe the shape of this distribution.
00:48
So i'm going to say that it is normally distributed.
01:02
And the reason we can say that is because we've got a sufficiently large sample size, maybe i should say approximately normally distributed, because our sample size is over 30, at which point you can start treating your sampling distribution as normal, regardless of the distribution of your population.
01:27
This is due to the central limit theorem.
01:29
So there's our answer for part a, that x -bar is approximately normally distributed.
01:36
Next, part b, we want to find the mean of the sampling distribution.
01:40
Well, it's a very general result that the mean of the sampling distribution is always just equal to the population mean mu.
01:49
So for us, that means our sampling distribution is going to have a mean of 50, since our population mean is 50.
02:01
So that's part b.
02:03
Next, for part c, we want the standard error of the sampling distribution.
02:08
So again, we have a general formula for that.
02:11
Our standard error, written sigma sub x -bar, is given by sigma, the population standard deviation, over root n, your sample size.
02:23
So that's going to be 10 for our population standard deviation, divided by the square root of 36, which is 6.
02:32
So 10 over 6.
02:35
So as a decimal, i guess i can see what that works out to be.
02:43
It's fine to leave it as a fraction, or even a reduced fraction, be 5 over 3 as a reduced fraction.
02:52
And as a decimal, to say four decimal places, it will be 1 .6667.
02:59
It's really 1 .6 repeating, but if you want to round, that's how we'll round.
03:08
So there's our answer for part c.
03:12
Next, what do we want for part d? now we want to compute some probabilities.
03:21
Okay, so let me make some room for computing these probabilities.
03:24
I'm just going to keep all of this information off to the side, over here, let's say.
03:30
And we'll also draw ourselves a normal distribution.
03:35
So we've got a visual aid while computing these probabilities.
03:39
So this is our distribution for x bar.
03:44
So we've got the mean in the middle, which is 50.
03:48
And let's see what we want to compute.
03:51
Well, for d, we want to compute that the sample mean will be between 45 and 55.
03:57
So that's the probability that x bar will be between 45 and 55.
04:06
So in terms of the area under the curve, this probability will be equal to this area, i'll now shade in red.
04:14
So 45 to the left of the mean is here, and then we want all the area up to 55, which is to the right of the mean...