 Hi folks, this is Dr. Don. I have a question out of Chapter 8, Section 1. These are two sample tests, and in this particular problem we're given some summary data, and I notice over here that we're given the population standard deviations sigma 1 and sigma 2. So that's a clue. Whenever you have sigma, the population standard deviation, whether it's a one sample or a two sample, you use the Z test. And here we're going to use the Z test to see if the population means are different. The question is, is the difference between the salaries in Region 1 and Region 2 of statisticians more than 5,000? We're getting this random sample, and this is the summary data, alpha is equal to 0.1. First thing we need to do, of course, is set up the null and alternative. Here we've got a more than, which is a greater than symbol, which means the alternative is mu 1 minus mu 2, the difference is greater than 5,000. The null has to be the complement of that, which is less than or equal 5,000. Because the wording here is, is it more? That tells me that in this particular problem, the claim has to be the alternative, because they're asking the question, is it greater than 5,000? We're going to solve this using the Excel calculator on my website, because they'll do it very, very quickly and easily. To find the calculators, if you're not using my video matrix, just go to my website, drdonright.com, click on Business 233, it'll open up this list. We want the two sample Z-Tests for difference between main and sigma known. So I'm going to click on that, and they'll bring up the calculator. I'm just going to roll this down until the calculator pops up. There it is. So now we just need to enter our data here. Our X1 bar is 68,500. Enter that. Our X2 bar, 64,000. Enter that. Our sigma 1 is 8850. Our sigma 2, 9150. Our N1 is 44. Our N2, 41. The difference is 5,000. Our alpha is equal to 0.1. That's okay. The next thing you do is select the claim operator, and as I mentioned, the claim operator contains a greater than. If it wasn't greater than, you just select on the drop down. That tells us that the alternative is the claim. This gives us the statement. The mu1 minus mu2 is less than or equal, greater than or equal, which is what the problem asks for. It tells us that the test is a right tail because the alternative claims contains the greater than operator, which points to the right, that's a right tail. Our test statistic is 4,500, which is just the difference in the two means. The standard area you don't need for this thing. The standardized test statistic is minus 0.256 or minus 0.26. Our p-value of obtaining that or more extreme value is 0.6, which tells us that the test is not significant. We would fail to reject the null, and therefore the claim is not supported. But the calculator will give us our critical value, 1.282. It tells us the rejection region is to the right of 1.282, and of course this test statistic doesn't fall in that region. Therefore, z is not in the rejection region, p is not less than alpha, that gives us a decision is failed to reject, and the conclusion is at the 10% significance level there's not enough evidence to support the claim. So this calculator is very, very quick. If you're given summary data, and I might mention that there's also a version in which you can key in raw data. That's a little more tedious if you've got a lot of data, but you can do it. And the calculator will help you avoid what I call the dumb mistakes by giving you all this information literally just by entering your data up in the blue cells and selecting the claim operator. So, hope this helps.