Example Two: A random sample of students was studied. Whether the student chose to sit in the front, middle, or back row of class was observed along with the person's GPA. Test the hypothesis that the mean GPA is not affected by row choice using a significance level of 0.05. Assume normality. Show all four steps. If statistically significant, find which row mean GPA is actually different. So lucky for you, you're actually not taking this class in person. Also lucky for you, just looking at this, it was a random sample of students, which is good, but the student is choosing where to sit—so this is an observational study. So no cause and effect. Last example was a controlled experiment. Doesn't necessarily have cause and effect though, still—just because small sample, we don't know about repeatability. But maybe worthwhile for you to kind of follow up for yourself for what should you should have. But for this one: observational study, so no cause and effect, even if we get statistically significant results, i.e., if we reject p-value—sorry, reject the null hypothesis because of the p-value. Okay, let's do this. Let's do this full test. We'll show you ANOVA and everything. Okay, so first step: Hypothesize Null hypothesis: There is no association. Or you can write mu1 = mu2 = mu3, right? All the means are equal. That’s a safe assumption, right? You don’t have different GPAs no matter where you sit—well, you don’t have statistically different GPAs no matter where you sit in the classroom. They're all about the same kind of differences, same kind of spread. Alternative hypothesis though: There might be a better row. Alternative hypothesis: One row has significantly higher or lower GPAs. Significance level. We were told right up here: significance level is 0.05. We'll assume CLT is fine—right? Said random, said assume normality. So CLT conditions are good. Now let's compute To help us compute—we need to put all of our data into lists in our calculator. Specifically, we're going to do an ANOVA test because our variable is GPA (numerical) and what sample we're in (categorical). Numerical, categorical: Anova. It'll spit out an F statistic. It'll give us a p-value. Realistically, on a quiz I might have you do full ANOVA and report out the F statistic and a p-value. But on a test, this is a lot of data to put into a calculator. This is probably one of those cases where I'll tell you to give me your hypotheses, I will give you these results (like we saw in that last example), and you'll just have to do the interpret step—so just the hypothesize and the interpret steps. All right, so switching my screen share. All right, let's clear this out—clear, clear, clear. All right, starting fresh. I go to STAT, ENTER, clearing my calculator. Unfortunately, did not clear out my lists, so let's do that real quick—scroll up, clear, enter, over, up, clear, enter, over, up, clear, enter. Okay, now my lists are clear and I can type in the front data. [typed in data from note sheet to L1.] That was just one list, and I can already see one error in my list. So again, good—you probably won’t have to do one of these from start to finish on the test. That's looking all right. Let me enter my middle data now. All right, same length [of list] of each—that looks good: 2.53, 2.65, 3.07, 2.92, 3.07, 2.1, 3.56, 3.62, 2.64, 2.87, right? Plus [List] three—I’m entering that data. If you haven’t finished typing in all of your data, that’s okay—just hit pause, type in your data, and hit unpause once you’ve typed it all in. Maybe even keep it paused, finish the whole analysis, and then check to see if we get the same results. Right, I’m hoping I typed everything in properly. Looking pretty good. I double-checked. I also have the results elsewhere so I can see if I get the same final results, and if not, I’ll come back and change the video. All right, so: STAT, TESTS. I want ANOVA. Scroll up—there it is, right at the bottom of all my lists. Hit enter. Unlike my other calculator functions, I have to tell it what list my data is in—so: second 1 [L1], comma, second 2 [L2], comma, second 3 [L3]. You can either hit enter or parenthesis, enter. There’s my One-way ANOVA. Bunch of data there, but I really just want the F statistic. I'm going to go to four decimal places [F=7.7780]—same for my p-value. p-value=0.0021—all right, that matches with my prior calculation. Go back to my notes. Alright, a lot of extra space here—ignoring that. ANOVA: F statistic, p-value. Based off of that and our alpha [0.05], what’s the decision we make? [writing decision and interpretation in notes.] Since we reject the null hypothesis, we have enough evidence for our alternative hypothesis. Enough evidence to say one row has significantly higher or lower GPA. Okay, so should we do a post hoc? Yes. We want to know—what’s the row to sit in? Which row has the higher GPA? I want to know where to maybe go sit next to people that can help me out in a class. Probably already have a guess. All right, we don’t have a post hoc procedure on our graphing calculator. We’ll just have to do a whole bunch of two-samp tests [error - should have been 2-SampTInt] —or List 1 vs. List 2, and List 2 vs. List 3, and List 1 vs. List 3. Okay, I’m going to do this on another screen and just kind of fill it in here because we did this back in 10.1. We've done a whole bunch of two-samp T tests [again, error - should have been 2-SampTInt]. I'm just going to write down my results. Okay, so I’m looking at these intervals. I’m checking to see if any of them capture zero or if they’re completely positive or completely negative. And we see that this one here captures zero—it was List 2 vs. List 3, so middle vs. back. So there's no significant difference between middle and back. But this is completely positive, and this is completely positive. So there is a significant difference when we take List 2 or List 3 and compare it to List 1. So the front does have significantly higher GPA than the middle or the back. Sorry—[mumbling]: front row has a significantly higher GPA than the middle or the back. How much higher? Well, front row has between—we estimate—between like 0.13 or so, between 0.1 or 1 GPA higher. Front vs. back: between 0.3 or 4 GPA higher. Now, just because it does say “significantly higher”—this is an observational study. There are definitely, probably going to be some confounding variables. In my experience—as both a teacher and a student—front students, they generally have to be there before class starts. Otherwise it’s a little bit harder to get to those seats. So it makes it easier to have a higher GPA if you’re there for the whole class, if you're there when it starts. Also, people in the front may be less likely to be distracted. They have to know, they have to be on top of their game. They have to be paying attention to the lecture. So that could also lead to some higher GPA. But not necessarily everyone has higher GPA [circled some of the raw data in the front row]. We see some lower GPA here. So it doesn’t take just sitting in the front, but maybe being able to be there for the whole lecture—not getting distracted—that makes it a little bit easier to have higher GPA. Also, if you’re one of those that sit in the back—I should also point out, when I go to department meetings or other meetings that I have to attend as a faculty, me and a bunch of other faculty are almost always sitting in the back. Back is nice. Back is more relaxed. I like sitting in the back. I like not having that constant attention. So it is fine if you're sitting in the front, middle, back—they all are fine. You can have a high GPA in any one of those seats. It’s just how much time you have to actually devote to your learning. All right, hopefully this was interesting to you. Let me know if you have any questions.