 Hi, one. I've looked at your homework for 8.2 and have some thoughts that may help. I don't know which technology you're using, so I'm going to try to show you both StatCrunch and Excel for these. The first problem, 8.2.9, you're given information that the null, the two means are equal alpha is 0.2 and the sample statistics are given, the summary statistics, the mean of the sample one, the standard deviation of sample one and the sample size. Same information for sample two. Note down here that you're told the variances are equal. Variances are just the square of the standard deviation, but that's an important factor. They give you the critical values, but the Excel calculator in particular will also give you the critical values in just one step. It's important when you're doing these things to pay attention to the two hypotheses here. We've got a statement of equality. That is the null hypothesis, but in some problems, they will make claims. You need to be aware of that. Down here in your answer, you got the test statistic correct, which is just the difference in the two means. X bar one minus X bar two is 3.8. The standardized test statistic, you got a three, and I think that may be because you didn't recognize that this is equal variances instead of unequal variances. Let's look at StatCrunch. I have StatCrunch open for all the hypothesis tests that we run in 233. You would start with Stat, and then you pick either Z-Stats or T-Stats for these differences of two means. Remember, you use T-Stats when you don't know the population standard deviation sigma. Here, we are not given the two population sigmas. We only have the standard deviations or the samples that we used to approximate them. We would go to T-Stats, and this is a two sample with summary. We bring up this dialogue box, and we just enter the data, and here I have the data entered. One thing you need to be aware of is this calculations options there. It says pool variances, which is not clear to a lot of folks. If the variances are equal, you need to pool the variances. They have a note here that this default was recently changed to off. In the past, and I think in some of my videos, the default was already checked. Here, if we have equal variances, we need to check the pool variances, and we need to set up the hypothesis test down here because the null was equal. That means the alternative must be not equal, which is okay. I'm going to go ahead and click compute, and we get this answer here, and you can see that's very much suspiciously like the answer you have here of 3.0, a little bit off. That was because we did it without pool variances, which is what you need for unequal variances. Let's go back, edit. This time, I want to check pool variances, and click compute, and we bring the information up there. You can see our t-stat 2.576 is correct. Our sample difference 3.8 is correct, and we've got a p-value of 0.0166, which is less than our alpha of 0.2, so that would tell you to reject the null. On this problem, we're given the two critical values, the lower critical value of minus 1.3, and then upper of plus 1.3. This test statistic, standardized test statistic 2.5676, is to the right of that, it falls into rejection, and that agrees with the p-value to reject this. I want to go back to the options edit for a second. Later on, one of the problems you missed, two of the problems, were for finding the confidence intervals for a mean difference, and you can get that very easily here at StatCrunch. Just check that, put in the confidence level you want, which is 0.2 for ours, and we click compute, and it gives us the confidence level, upper lower limit and upper limit. That, of course, matches the variances. This is equality. If I go back and edit and change that so we have unequal variances, compute will get a slightly different set of confidence intervals. So be aware that you can get that very quickly using StatCrunch.