00:01
So we have a consumer group that's interested in testing the automobile manufacturer's claim that a new economy model will travel at least 25 miles per gallon.
00:14
And so the alternative for this test would be a less than or a left tail test.
00:21
And so it says in part a with a .02 significance level, we have a sample of 30 cars, and it says what would be the rejection region for this? and assume that the standard deviation is 3 miles per gallon.
00:40
So our rejection region would be that region that is off to the left and has .02 down here.
00:47
And so that would be for, and we want the rejection rule based on the value of x bar.
00:55
So we want to know what that x bar would need to be.
00:58
Well, let's find what that z value is.
01:00
And so when we look up, we can either use the inverse normal button on our calculator, or we can look up in the table to find what that value is.
01:12
But that z value is approximately negative 2 .0537.
01:19
And so if we take our x bar and subtract away 2 .0537 standard deviations times the 3 divided by the square root of, and we had a sample size of 30, that will give us the x bar.
01:35
So i can use my x bar of, or the mean, i should say that 25.
01:43
So 25 minus, and i'll use that value or plus that value, and then times 3 divided by the square root of 30.
01:58
And so if our x, if we get a mean that is less than or equal to 23 .875 miles per gallon, that is where we will reject the null.
02:14
Now on part b, it says what is the probability of committing a type 2 error if the actual mileage, so the probability of failing to reject, if the actual mileage is 23 miles per gallon.
02:34
So now, because a type 2 error is where we fail to reject.
02:39
So this was the rejection region we want to fail to reject.
02:42
And so if that is the actual, then we convert it to a z value, 23 .875 minus the 23 over the 3 divided by the square root of 30.
02:55
And left parenthesis, 23 .875 minus 23 divided by left parenthesis 3 divided by the square root of 30...