 Hi, this is Dr. Don. I have a problem from Chapter 8, Section 2, which is on two sample tests. And if we read this problem, we're told that an engineer wants to know if producing metal bars using a new experimental treatment rather than a conventional treatment makes a difference in the strength of the bars. They give us an alpha, and then we've got several questions we have to answer. They tell us the population variances are equal and the samples are random. And they hit us with, if convenient, use technology to solve the problem. And that's what we're going to do. But first, let's answer as many of these questions as we can. The first question is, identify the claim, state the null, and the alternative. Well, this is kind of a fuzzy question. It doesn't say specifically, I don't see the word claim in there anywhere. So that's kind of fuzzy. But the question is, the engineer wants to know if producing metal bars makes a difference. Okay, remember that our null hypothesis for all of these tests is no difference. That's the null. And in this case, if he's wondering, if he wants to know, he's wanting the test, if it makes a difference, then somebody, perhaps himself, thinks it does make a difference. So that would be the claim that the treatment does make a difference. You wouldn't be developing an experimental treatment if you didn't think it was going to help, would you? So the claim is that there is a difference. The null then has to be no difference. So we go down here, the claim is the new treatment makes a difference. The null then has to be that mu one, the mean tensile strength of the old, is equal to the mean tensile strength of the new, and the alternative is, of course, not equal. And reiterating there, the alternative is the claim. Remember because we have a not equal sign in the claim, that means that the critical value, excuse me, that it's a two-tail test, and that we have two critical values. Now this student got the wrong answer there, and when I dug into what she was doing, she didn't recognize that we didn't have sigmas up here. I don't see any sigmas. The population standard deviations given. She mentions the population variances are equal, but it doesn't tell us what the population variances are, or what the population standard deviations might be. If we know the variances, of course we can take the square root and get the population sigmas, but here we don't know anything other than to assume that they're equal. If we don't know the sigmas, then we must use the t-test. And I will show you in a minute, once we get StatCrunch open, how to find the correct values of critical value of t using the right degrees of freedom for a t-test for a two-sample, which depends upon whether or not we have equal variance assumption or not equal variance assumption. Let's open this in StatCrunch. So we have our data there in StatCrunch. Okay, let's get our critical values first. We're going to go to Stat, Calculators, t, open up the t-calculator. Now we need the degree of freedom, and because our variance are equal, we use the formula that the degrees of freedom is equal to n1 plus n2 minus 2. And here we have seven values of experimental, ten, that's n1, ten values for conventional, so that would be 17 minus 2 or 15 for the degrees of freedom. This is a two-tail test, therefore we put half of alpha, and alpha if I remember correctly is 0.01, so we put 0.005 and click Compute, and we get a critical value on our low side of minus 2.467, and of course by symmetry we know that there's also a positive value 2.947 on the top side of this. So our two values are minus 2.947 and plus 2.947. Okay, let's run the hypothesis test. We go to Stat, t-stat, two-sample with data, open up our dialog box, the two-sample t-test. Our values for sample one are in the experimental column. Our values for sample two are the conventional column. This is a critical step here. If we've got equal variances, then we must pool the variances, and so this box needs to be checked. If you were given in your problem to assume the variances were not equal, we do not want pool variances, so we would uncheck that. But here we have equal variance, so we check that. Our hypothesis test, mu one minus mu two, the null this time is zero, just no difference, and that means that the operator and the claim has to be not equal. And that's really all we need. We just click compute, and we get the results here. We get our sample difference of mu one minus mu two, which in some questions they will ask you for the test statistic, which is just mu one minus mu two. Over here we've got our standardized test statistic, t-stat of 1.15 and around that be 1.151. We also have a p-value of 0.2678, which should tell you because the p-value is much larger than our alpha of 0.01 that we would fail to reject the null. But also in this case we're going to look back at the, compare the t-stat to the rejection regions and you should have the same answer. If you remember correctly the, I'm going to bring that up here. You've got our critical value on the low side of minus 2.947 and the t-standardized test statistic does not fall in there. And of course on the upper side by symmetry plus 2.947 and 1.15 is back in here somewhere. So we're definitely not in the rejection region. So that would be your decision. No to reject.