 Okay folks, this is Don. I have a problem that some of you are still struggling with, and I want to go through it and show you again how you can solve and find all of the requirements for this type of problem. One of the limitations of stat crunch is that if you are doing a paired samples hypothesis test, which this is a paired samples, and if you are not given the raw data, if you're just given the summary data, then it's difficult to calculate this standard deviation of the differences. But if you are given the raw data, then all of these values you can calculate very, very quickly. Let's start again by identifying the null and alternative. You have to get that right in order to be able to come with it, to run the test right in stat crunch or Excel or anywhere, and also to be able to come up with the correct conclusion. And the claim is that is there enough evidence to conclude the fuel additive improved gas mileage. And so the claim is that the fuel additive improved gas mileage. Alright, if it improved, that means that the mileage after using the additive was greater than the mileage before using the additive. That means that there is a less than operator in the claim. And when we look at the alternatives here for the null and the, I'm sorry, looked at the options here for the null and the alternative, you look at this one, the difference has to be less than zero because mu two after the additive is greater than mu one. So that has to be the alternative, which is the claim. The null is always an equality, which means it's greater than or equal, equal less than or equal. So those are our hypothesis. Let's find the critical value. We've got an alpha point one. And I'm going to go ahead and just load up this data and stat crunch. Okay, we've got our data. Right now we want to use the calculator. And I have to ask you a question. Are we going to use the normal distribution or the t distribution? Well, for paired samples, we always use the t distribution. And particularly, in this case, where we've only got an N, the number of pairs of eight, much less than 30, that should tell you also that we use the calculator for the T. So here's our calculator. We need to put in the degrees of freedom. We've got eight pairs. Therefore, the degrees of freedom is seven. And alpha is point one. We have the alternative is pointing to the left. That means we've got a lower tail test, left tail test. So I'm going to leave that pointing that way. And we're going to put in point one, because the entire area that we're interested in is in that left tail. And we get a critical value of t t sub zero of minus 1.415. And that is the answer that we need. The rejection region, of course, any t that we calculate the standardized test statistic t that is less than minus 1.415 would fall in the rejection region. And therefore, we would reject the null. Okay, let's go ahead and calculate the rest of the data. We need the D bar, which is the mean of the differences, the standard deviation of the differences, and our test statistic. And we can get that and stat crunch just by running our t stats paired sample. And we always have to have the data. So we've got mu one is car mileage without additive, mu two with additive, go ahead and click the differences and I'll show you why that will save you some time. Now we've got to set up our hypothesis. Remember that in stat crunch and other statistic software, we only test for the equality for the null. And if we've got a greater than or equal, if the quality is significant, then the greater than portion will also be significant. Our alternative has the less than operator. And I'm going to just for the hunt of click on, give me the summary statistics and click on compute. And here we've got our data. And we've got most of what we need. We've got our mean. And this is the main difference D bar minus point two seven five. And that is just this difference in these these means minus point two seven five. And that is the answer in my stat lab. We've got our test statistic minus seven point two nine. And over there in my stat lab, that is the test statistic they were looking for point seven to nine. So just by running the hypothesis test, we get our two of our data points, the D bar, which is known as the mean, and our T stat, which is our test statistic, T test statistic. Okay, our P value is point two four four nine. And that tells us that we do not. We failed to reject the null hypothesis because the P value is much larger than the alpha point one. And we can also see that our T stat of minus point seven two nine is not greater than or is does not fall into the rejection area. Let me just call that rejection area back up again. I don't why close it out. But we've got seven and point one. Okay, there's our rejection region anything smaller than minus one point four one five. And of course, our T stat minus point seven two eight is in here somewhere. So it's definitely not in the rejection area. And that again is supported by the P value being less than I mean, much greater than the alpha value. Okay, final piece of information we need is s sub D, the standard deviation of the differences. Now you recall when we were running the hypothesis test, I said check that box that saves the differences. Here are differences. And now all we need is a standard deviation of that. And we can get that very quickly with summary stats, columns, and I want to go on the differences. And all I really need is a standard deviation compute. And we get a standard deviation f sub D of one point zero six seven, which is the answer in the test. So I hope you you learn to use your technology tools to help you get through. And I hope this helps. I'm not going to answer the the rest of the problem because we've been through that before. So hope this helps.