00:02
This problem says that a bank loans officer rates applicants for their credit.
00:06
It says that these ratings are normally distributed with a mean rating of 200 and a standard deviation of 50.
00:13
It says, suppose that we draw 40 different applicants at random, so this is the sample size n, which is 40.
00:22
And part a asks us to find the probability of the average or their mean being above 220.
00:28
So what we're finding here is the probability of x bar, the sample mean, being greater than 220.
00:34
Now before we do anything, we'll have to recognize that we're dealing with a sample, and since that sample size is greater than or equal to 30, that means that we can find our mean for the sampling distribution as well as our standard deviation for the sampling distribution.
00:48
The mean of the sampling distribution is equal to the population mean, so that is just 200, since we know our population means 200.
00:57
However, our standard deviation for the sampling distribution is equal to our population standard deviation divided by the square root of the sample size n.
01:06
So we take our population standard deviation, which is 50, and we divide it by the square root of our sample size, which is 40, and that gives us a sampling standard deviation of approximately 7 .91.
01:20
So these are the mean and standard deviation that we use for this problem, since we're dealing with the sample.
01:25
We're now ready to find our z score for 220 using this red z equation that i have up here on the screen.
01:31
Using the equation z is equal to x, which is 220, minus the mean, which is 200, divided by the standard deviation, which is 7 .91, and then if you solve that in round for round to two decimal places, you should come out with 2 .53.
01:50
That's our z score.
01:52
So now we can turn to our z table that i have on the left side of the screen and find it.
01:56
We'll go down to the left -hand column to 2 .5, which is here, over to 0 .03, which is up here.
02:02
When we meet in the middle then, we see that we get 0 .9943, and remember our z table always gives us what's less than, but since we want what's greater than here, we want what's greater than, we'll have to subtract this from one.
02:19
So if we take 1 minus 0 .9943, we'll get our answer, which comes out to 0 .0057.
02:29
So you can see there's a very, very small percent chance, less than a one percent chance, such as 0 .57 percent of the sample mean here being greater than 220...