 Hi, this is Dr. Don. I'm back with part two of that problem out of chapter nine on a two-way ANOVA, a two by two factorial experiment with two observations, which is recall, which means this is an ANOVA with replication. If you recall, we used that crunch after we rearranged the data and did the two-way analysis of variance and got our ANOVA table and our MEANS table to answer the first part of the question. Here is the ANOVA table and here are the answers we needed for that. As we scroll down into part C, it says, test to determine whether the MEANS differ at alpha .05. Does the test support your visual interpretation? And you recall, visually we saw that there was a difference between MEANS, but there was not an interaction and the null and alternative for this test is there are no differences, the no difference null as usual, and at least two MEANS differ for the alternative. Now the way that we're doing this in this particular problem is a little bit backwards. If you think about it, look over here on our ANOVA. If we've got a significant p-value here and a significant p-value there, both less than .05, even though the interaction is not significant, the p-value there which jives with the graph where the two lines intersect, we know that we've got something significant going over here. So we don't really need this test to see if the overall ANOVA is significant. It will be. But let's get them the answers they need. They want to know the test statistic, f sub t, and they want to know the p-value. So to get that, StatCrunch doesn't give it to us, we're going to have to get in to Excel. So let me show you the Excel. Here's the calculator. I'll have this available so that you can download from my website. The way it's set up is just to use the equations that would calculate this f sub t, our test value, test statistic of f, and then the p-value associated with that, the two answers we need, and there's some formulas that you have to use to get there. The way I set it up, I've got a ANOVA blank up here, and the blue cells are the things that I need to copy into. And it works for both the StatCrunch version of ANOVA as well as the Excel data analytics ANOVA that I'll show you in another video to copy into these blue box. The critical cells are these that I've got highlighted there, the mean square cells for the sample, your row variable, B, your column variable, your interaction, and then the error. Those four values are the critical things we need for this question C. If we go back over here into StatCrunch, there's not an easy way to get this information down. The way that I found that works for me is to just take my cursor and highlight that part that I need, use CTRL C to copy it, then I'm just going to go someplace in my Excel here and then right click and then paste. And you can see that we get a very small version of that. I'm going to go ahead and highlight this, make it a little bit bigger so you can see, go back up to 11, so we can see those values there. Now the values that we need, we'll eventually need to use all of those. Looking at these, A, B, interaction, I've already got those. The degrees of freedom just happened to be the same, so I don't have to change those this time. My sum of the squares, I'm going to copy the first four there, CTRL C, and paste those there. I'm going to get the final version of my total, CTRL C, paste it there, and then I'm going to get my mean squares. These are really the most critical part, CTRL C, paste them there, and then I'm going to get my F values and my P values, CTRL C, and paste them there. And we'll use these things for the answers that are to come. Now the only else you need to do is to go into the second part of the blue cells, enter your alpha value, which is .05 in this case, our K, which is the number of treatments, or our A factor was two, remember that, our column factor, our blocks was two, and the number of replications we had, remember we had two observations per treatment, so r is equal to two. Enter those values, and then all of the yellow cells downstream propagate, and down at the bottom it will check to see if P is left in alpha in this case. Some problems we'll ask for the critical values, so I've given you that as well as the P value, and in both of those, in this case, say reject the null, because either the F test value is in the rejection region, or P is less than alpha. Well, let's go back and check. Here's our answer, 22.414 and .006. So I'm going to pause and go back into my stat, okay, we're in my stat lab again, and here's our part C, the test statistic 22.414, which is the answer that the calculator gives you, the P value, .006, gives you that answer there. So let's scroll on down, does the test support your visual interpretation? Yes. We reject the null, down here we reject the null. There is sufficient evidence that at least two of the treatments, in other words, we reject the null, that means the alternative, which is at least two of the treatments differ at alpha, .05, which agrees with the graphs that we plotted. Parts D and E are similar, it's asking us to make a determination whether or not the main effects and the interaction in the experiment were significant, and it goes through the same process, asking you to get the null and alternative, I'm not going to spend time on that, you can do that, I'm going to show you how we get the statistics they need. First it says, find the test statistic for the interaction. And here we look in our ANOVA output, there's our interaction row, we read across there, the test statistic is .156 rounding, and the probability as we said before is .713. So that's the answer to that part. We're going to scroll down here into E and it asks about the two treatment effects, and the first one it says what about the effect of factor A, and factor A if we look here in our row, factor A, and the F statistic is 48.728, and the p-value is .002, which matches their answer there. The next part asks about factor B, and we get the answer there, the F statistic for factor B, which is our column blocking variable, 18.358 and .0128. So that's how you get the statistics you need, the rest of the questions, I think you know the logic of how to say whether or not the p-value tells you to reject or fail to reject the null. So I hope this helps you see how to get the statistics part of this long problem.