 Hi, this is Dr. Don. I have a problem out of McClave in my stat lab. This is problem A.2.3T and it has to do with a two sample test for population means. Now, one of the things you need to pay attention to, particularly if you took statistics 233 from Excelsior, different authors have different ways of determining which test you should run. McClave uses the discriminator of the sample size. If the sample size is greater than 400, McClave says you should run a Z test even if you don't know the population standard deviations. Or in this case, we don't know the sigmas, we have the S's. So using the McClave procedure, we should use the Z test, the two samples Z test. They want us to give them the 95% confidence interval on the difference between the means, which is the test statistic, and then interpret that interval and then run a hypothesis test. We're going to do this using the Excel calculators on my website. Remember, if you go to my website, it's www.drdonwright.com and when you get there, click on the business 233 and 503 link. This page will come up that will list a number of calculators I have here for confidence intervals and hypothesis test. We look down, we want a two sample Z test for difference between means. It says sigma known. In this case, we don't know sigma, but again, McClave, unlike Larson, the 233 author, says if an N is greater than 30, you can use the Z test. So we're going to use the Z test. I'm going to click on that and the page comes up and it gives you information. Again, it says enter your data and the blue cells only, select the math operator and the claim by clicking the orange cell and then all the yellow cells update. So let's drop down here a bit and here's our calculator. We need to enter the information in blue. We need the sample one main, which is 5293. I'll hit enter. Sample two main, 52, 58, hit enter. Sample one standard deviation, or S in this case, 149, enter, 203 for sample two, we have hit enter 400 for both samples. Our main difference in this case, looking over here, is zero. So we put zero in the main difference. Our alpha, if we look down here, is 95% or an alpha 0.05. So I'm going to put 0.05 in there and that gives us the first part of the test done. Now we need to enter the claim operator. In this part B, the claim is the alternative in the McClave way of thinking about things. So it's not equal. So I click in my orange cell to get the drop down box and I find the not equal sign. Then it will give you statements of the null and alternative. The format they're using here is mu one minus mu two equals zero. And the alternative is mu one minus mu two is not equal to zero, which matches that. Our test statistic and here I differ a bit with the McClave. Most authors call the test statistic, which is the difference between the two. In this case is 35. The standardized test statistic, once you go through and convert that to a Z score, is a standardized test statistic of 2.78. But that's what they're asking for here, 2.78. And if you happen to miss that on the test, I will give you credit because if you put in the correct Z score there instead of the test statistic, we want the observed p value. We look down here and it gives us a p value of 0.005, which is the p value they want there. And if we look down to the bottom it says we want to reject the null at the 5% level. There is enough evidence support to claim that the difference between the two means is not equal to zero. And that's what this says here. Reject the null, there's sufficient evidence, population means are different. Let's go on down to part C here and it says suppose the test in part B was conducted with the alternative hypothesis greater than zero. How would your answer change? All we need to do is to go into our calculator because all the data stays the same. Let's change the math operator to greater than to match that and we can double check mu 1 minus mu 2 is greater than. That matches the alternative. The observed significance level now drops down to 0.003, which is what they want there. So that gives us that particular answer. And again, if we've dropped all the way down, it would give us a conclusion that we would reject the null again. Part D says test the null hypothesis mu 1 minus mu 2 equal to 21 versus the claim, the alternative not equal to 21. So all we need to do is go up here to our main difference, put 21, and then we want to change the claim operator again to not equal. And it may jump out of the screen there a little bit. So now we can see that matches the alternative is mu 1 minus mu 2 is not equal. Our test statistic standardized is 1.112 and our p value is 0.266. And the conclusion is do not reject the null because our p value is above 0.05. And that's what the calculator tells us fail to reject the null. We do not reject the null. At the 5% significance level, there is not enough evidence support to claim. There is not enough evidence to include that mu minus mu 1 is not equal to 21, which is the claim. And then we compare the answers. The test in part B supported the hypothesis of the major difference. The test in part D supported the hypothesis of the main difference is 21. Because again, we're saying if it's not different, we do not support the claim. That means the null that the main difference is equal to 21 is correct. And then finally, what assumptions we must assume that the two samples are independent random samples. We don't need to assume they're too small. We don't assume the Z distribution is poxly normal. The Z distribution by definition is normal. And we don't need to assume the two samples are dependent because that wasn't worked. They've got to be independent for these tests. So I hope this helps you see how you can use these calculators on my website to really help you get all the way through the hypothesis test and the confidence interval without making a bunch of dum-dums. So I hope this helps.