So let's assume you've done ANOVA, you've done your analysis, and you get to the end, you're like, "Okay, there is an association." But again, we talked about how ANOVA doesn't tell you where the association is. That's what a post hoc procedure is supposed to do. So post hoc is just Latin for "after this." As I explained here, post hoc—we would perform the post hoc procedure whenever we reject the null hypothesis, aka when we have statistically significant results. Some statistics programs do have features built in to do this, like we're going to see below in Example One. Otherwise, we would just calculate confidence intervals. Specifically, two-samp T-Int calculations, right? Let's practice this. Example One A researcher predicts that students will learn most effectively with a constant background sound as opposed to an unpredictable sound or no sound at all. The researcher randomly divided the 24 students into three groups of eight, and students study a passage of text for 30 minutes with the appropriate background noise. After studying, all students take a 10-point multiple choice test over the material. Analyze the following results. So here are the results. When we look at this, we see that one variable could be what group they're in—that's the categorical part, the samples—but our actual overwhelming variable is the results from this multiple choice test. So it makes sense: this is definitely an ANOVA. All right, let's state our null and alternative hypotheses—H-naught and H-a. Null hypothesis: There's no association. Or another way to write this would be like, all the mu's are equal. Alternative hypothesis though: One of them is significantly better or worse than the others. So: alternative hypothesis, students learn significantly better or worse with one of these specific sounds. We're assuming CLT is fine. We're already given the calculator output. So, yeah, this was the raw data. If you want to, you could put this data into, like, List 1, List 2, List 3 and run an ANOVA test. Right, generally when you run the ANOVA test, you would get something like this [circled the ANOVA table]. As we already saw, we are really just interested in these two pieces of information here [circled the F-Stat and p-value]. But it is kind of nice to have these summary stats. And the feature that goes along with our book—it's called StatCrunch—its option to analyze and do the post hoc is this: Tukey HSD results. So this is the post hoc built into StatCrunch for us. This is very similar to if we calculate those two-sample intervals. All right, lots of information here. Let's look at the summary stats a little bit more real quick. Just looking at these again—we knew 24 students were split up equally into these three groups. Looking at these means, we can already see constant sound was a higher mean than random sound or no sound. Interesting. But even though there's difference here, is it statistically significant? Right, okay. So ANOVA compared all of these all at once, reported this one p-value [p-value = 0.0454]. If we compare this to, say, alpha = 0.05—which they did not give us—assuming we used our default significance level, this p-value is less. So we would do what? Since our p-value (0.0454) is less than 0.05, we would reject the null hypothesis. So we have enough evidence to say that our alternative hypothesis is true [copied and pasted the alternative hypothesis sentence in context from above]. But, considering you are students, you're probably wondering, well, what sound is better? What is that better sound to have? Now, looking at our summary stats, you can kind of guess—it’s better to have constant sound. Actually, not random sound, not no sound, but actually constant sound. But maybe even random sound is better than no sound. Let's go look. So looking at this data here—these are again similar to the two-sample interval—it's saying what the difference was for the sample, and then lower and upper bounds. This is supposed to be your interval. Or if you did a hypothesis test, here’s a p-value. So the part that is similar to a two-sample T interval are these parts here. Now, one thing I do not like about this is the way it’s phrased: "constant sound subtracted from random sound" or "constant sound subtracted from no sound." That interval—it’s not phrased in the best way. But looking at all these intervals, we want to identify the interval that either captures zero—so there's no difference—or is fully positive, in which case group one is significantly better (larger), or fully negative, so the second one is significantly better. All right, so this first interval: negative to positive — so, no difference. This one: fully negative [middle interval]— [last interval] negative to positive — so, no difference. All right, here we have our difference. This is constant sound versus no sound. Again, it's hard based off of this sentence. I find it difficult to figure out which group is actually group one or group two. So I just know that we're comparing constant sound and no sound, and I can just use my sample data to tell me—Okay--people that had constant sound do have significantly higher scores than the people with no sound. This was a controlled experiment. These results stand up a little bit more—they're a little more helpful, a little stronger. But of course, it's just one sample. These are sample sizes of eight. We want to see if this repeats and gets similar results. But it's interesting. Since you are doing quizzes at home, maybe you can test this out for yourself. Would it be better if you had, say, constant sound—maybe like a white noise or brown noise or pink noise type sound in the background? I'm not saying you have to have like classical music, but maybe that's better than having random sound or no sound. Let’s do one more example...[lead into next video]