 Hi, this is Dr. Don. I have a problem out of Chapter 8, Section 2, on a two-sample test of the mean difference. And we are to get both the confidence interval around the mean difference as well as do some hypothesis tests. We're given some sample data here, but we're told that we have independent random samples of 400 observations each. Unlike 233 and the Larson's text, where the determinant, whether or not you use a z-test or a t-test, is knowledge of the population's sigmas. Here we don't know the sigmas, therefore we would use the t-test and the Larson 233 course. But in McClave and 503, the discriminator we have to use is whether or not the m is greater than 30. And here it's much greater, it's 400, so we use the z-test. Just remember that. In McClave and 503, you look at the sample size, and if it's greater than 30, then you use the z-test unless you're told otherwise. Okay, we're going to do this using stat crunch. Remember, you can go to question help and then open up stat crunch. I already have stat crunch open over here to have a little bit of time. And what we need to do here is go to stat, z-stats, because we've had z-stats, we have two samples with summary. And we bring up the dialog box. We need to put the mean in there, 5312, the standard deviation of the first one, 160, the sample size 400, and then again 5276, 190, and 400. The first question is about the 95% confidence interval. So we go down here and we click on that radio button to say we want the confidence interval for the mean difference, mu1 minus mu2, 95%. And we click compute and we bring up a answer box. We've got our sample mean. This is the difference. That's 36 mu1 minus mu2. We've got a lower limit of 11.65. I think they want just to one decimal point, so that would round to 11.7. And then the upper of 60.3. And that gives us the answer we need there. Okay. And of course, the interpretation of that is we are 95% confident that the difference, this 36, falls in that confidence interval. So that's pretty straightforward. Part B says, test the null hypothesis, mu1 minus mu2 equals zero versus the alternative, mu1 minus mu2 equals zero. So all we need to do, we don't have to reenter the data. We just click on options, edit, and it brings up our dialog box. This time we check we want the hypothesis test for the mean difference. The difference for the null is zero. And the alternative has a not equal operator. So we select the not equal operator. And we click compute. And we get our answers here. We get that the z statistic, which is also known as the standardized test statistic by most authors. Over here in the homework, though, it asks what is the test statistic. And since they give you the z, that's a clue. They want the z statistic, or the standardized test statistic. We have 2.898, which rounds to 2.90, which is the answer they want there. They want the p value given here 0.0037, which would round to three decimal places to 0.004. So that is pretty straightforward. We interpret the results because the p value 0.003 is less than alpha of 0.05. That means we reject the null. And that tells us that there is sufficient evidence that the population means our difference. Remember, if we reject the null, we accept the alternative. And here the alternative is that the means are different, that the difference is not equal to zero. So that's how we answer that problem. Let's go to part c. Suppose the test in part b was conducted with the alternative mu 1 minus mu 2 greater than zero. All we need to do again is go to options, edit. And then we change our operator to greater than to match that. And we click compute. And we get our answer box again here. Our test statistic didn't change, but our p value did. And it is 0.002 rounded to three places. So, and again, we would say reject the null because the p value is less than alpha 0.05. And we would accept the alternative that mu 1 minus mu 2, that difference is greater than zero, which it is sample mean here the difference 36 is greater than zero. Okay, D test the null hypothesis mu 1 minus mu 2 equal 22 versus the alternative not equal to 22. And again, all we need to do is go to edit because our data doesn't change. And we change our operator here to not equal. And we change our null hypothesis to 22. And we click compute. And so we've got our answers up here that the test statistic now has changed because we changed the null value to 22. It's 1.13, which is the answer they want over here. And the p value is 0.260 rounding the three decimal places. And we go down to the bottom there again because the p value is greater than alpha this time, we do not reject the null. And therefore, we say there's not sufficient evidence to conclude that the mean difference is not equal to 22. In other words, the alternative was that it was not equal. And we can't support that with the evidence. Okay, the last part is compare your answer to test conducted in part B, choose the correct answers. And if you go through those one by one, you'll see the test in part B supported the hypothesis because we rejected the null and the test in part D supported the hypothesis that the difference is 22. It's we fail to reject the null that the difference was 22. And again, the final part is the assumptions is that the two samples are independent random samples, which is what we've got up here. So I hope this helps.