 Hi, this is Dr. Don. I have a problem out of McClay Chapter 7, Section 0.4, and it is about doing a test of an hypothesis. And if we read this, it says a random sample of 100 observations from a population with a standard deviation of 34 yielded a mean. Okay, right there, critical to analysis. This is a single sample. It's not two samples. The population standard deviation, sigma, is given. That tells you that we're going to use a Z test. And we're given some questions to answer, finding the test statistic and the p-value, and deciding whether or not to reject or not to reject the null hypothesis. I have put together some Excel-based calculators on my website that you're free to use that go through this process and would really save you a lot of headaches. So let's bring that up. If you go to my website, just www.drdonwright.com, go click for your Business 233 right now and I'll change this to make it easier to find it. But click there and it will bring up a page that has some links to some calculators on here. And if we look down here, Excel calculators for hypothesis tests, single sample, Z test for population mean, sigma known. That's what we want. So I'm going to click on that one and it brings up this page, takes a second to load. Okay. Here we have the information and I mean the calculator and it says to enter your data in the blue area and then select the claim math operator. That's a key point. And then the calculator will update all of the yellow cells to give you the answers you need for almost any question you'll run into in my stat lab. Okay. So the first thing we need is I want to drag this down so we can see a little better there. We've got population mean that our null hypothesis, mu is equal to 100. The population standard deviation is 34. The sample mean, 106. The significance level is, let's see what are they asking for there, .05. And the sample size is 100. We have the blue data in and now we need to select the math operator to put in this orange cell. Now the calculator now says what is the claim operator. So let's inspect the hypothesis over here in my stat lab. It says test the null hypothesis, mu equal 100 against the alternative that mu is greater than 100. You recall the null always has to be a form of equality and here they say it's equals. The alternative has to be the complement of the null. In this case, the complement could either be greater than 100 or less than 100. So that tells me even though they're not saying that the alternative is the claim, the fact that it's got a greater than operator tells me that we should consider the alternative to be the claim in this particular problem. Okay, we go over into the calculator, click in the orange cell, get the drop down and we find the greater than operator which matches our alternative over here. And we can ignore this part. This particular question doesn't ask that. What is important that this is an upper tail, right tail test and that gives us a Z value of 1.765 or 1.76. We go over here, that's what they want, 1.76 which matches the right tail, it's positive. The P value down here is 0.038 or that rounds to 0.039. They want three decimal places which is the answer they want there. The final part says interpret the results. We drop down to the bottom of the calculator. It tells us that it gives us the critical value which our problem doesn't ask for. It gives us the rejection region which is the right tail which is what we would imagine. And is Z in the rejection region, yes. The Z of 1.76 is greater than 1.645 so it's in the right tail. That tells us that we would reject the null. And P is less than alpha, 0.038 is less than 0.05 so there again we reject the null. And this calculator tells us that. At the 5% significance level, there is enough evidence to support the claim that the mean is greater than 100. If we go over here, it says reject the null. There is sufficient evidence which matches that, that the true population mean is greater than 100. We have to go back and interpret this again. Our alternative was greater than 100. We're rejecting the null therefore we accept the alternative. So let's look at the next part. It wants us to test the null with the same data that mu is equal to 100 against the alternative hypothesis, mu is not equal to 100. So we go back into our calculator and all we have to do is click in the orange box again and we want not equal. Our alternative has the not equal operator. And so that changes into a two tail test. The Z value, the test statistic doesn't change of course because our data hasn't changed. But our P value does change because we now have a two tail test. And if you remember from the math that in the two tail test we have to multiply the P value that we get out of the tables by two. And here this calculator does that for you. The answers are again that the P value is not less than alpha therefore we fail to reject. And the calculator will give you the critical value and the fact that this test statistic does not fall in the rejection region. So both of those say no, fail to reject. And the conclusion is there is insufficient evidence to indicate the population mean is not equal to zero, which is what we're saying there again. There is not enough evidence to support the claim that the true population mean is not equal to zero. Final question is compare the results of the two tests and explain why the results differ. And the answer is results differ because the alternative in part A is more specific. In part A we said we think that the mean is greater than 100, that's one direction. In part B we said not equal, which is two directions so it could be either less than or greater than is just not equal to. So that is less specific. Check out these calculators on my website. They may help you avoid what I call dumb dumb errors when you're doing hypothesis tests.