Example One: Random samples of gasoline prices were obtained in three cities and are shown in the following table. To make sure that we didn't have to input a lot of data, there's only sample sizes of five for each of these cities. If we had more time, energy, and motivation, it would be a good idea to have more data, but we're just going to work with five for each of our samples. But we've got three samples here. First question: how many comparisons would have to be made? So you could just be remembering from the prior page—three samples, three comparisons. You know how many comparisons those are. But I'll go ahead and actually list them this time. So we should do like Denver versus Houston, Houston versus Cleveland, and Denver versus Cleveland. So three comparisons. Assuming the overall significance level is 0.05, what is the Bonferroni-corrected level of significance for each pair of comparisons? So we're just going to compare two at a time, take our overall significance level, divide by the number of comparisons, and we should get 0.0167. All right. Now let's go ahead and start analyzing this data. Okay. To help, it would be nice to know: what are the sample means for each of these samples? And then we should carry out our t-test comparing the means of all pairs assuming the CLT holds. All right. To hopefully maximize the amount of time I'm sharing my screen for my calculator, let's maybe set up my t-test. Just doing my null hypothesis. So what is the safe assumption? Each of my t-tests will just be a two-sample t-test. We're going to assume the CLT is good. We'll do a two-sample t-test comparing each of our proportions. In general, because we're looking at two samples, what is our null hypothesis? So null hypothesis is always equals. So null hypothesis is that mu1 equals mu 2 equals mu 3—all the means are equal—even though we'll only be working with two at a time. So maybe mu1 equals mu 2, mu 2 equals mu 3, and mu 1 equals mu 3. Alternative hypothesis—we're comparing—doesn't necessarily say that one is bigger or smaller than the other. So our alternative hypothesis will just be the "not equal to" option. And then we'll go ahead and carry out this two-sample t-test for each of these comparisons. When we do our test, we do want to list what is the test statistic (the t value) and what is the p-value. All right, let's go ahead. Let's take this data. We're going to put these in some lists. Probably use List 1 for Denver, List 2 for Houston, List 3 for Cleveland. So when I calculate my sample data, this will be Denver, List One. x1-bar is about... and Houston, List Two. x2-bar is about... and Cleveland, List Three, x3-bar is about... switching my shares. All right. Here's my graphing calculator. Put my data into a list. I'm going to hit Stat → Enter. I already have data in this list. Let's scroll up, Clear, Enter, Over, Up, Clear, Enter, Over, Up, Clear, Enter. Okay. So List One should be my Denver data. I'm just glancing back. Also, I think these days is definitely older data. I don't remember when gas prices were under $3. Houston data. And Cleveland. All right. We can calculate the sample mean for each of these by doing 1-Var Stats for each list. That is one route. Also, later, when we do our two-sample t-test, it will report my sample mean for each one. So I could go the shorter route and just jump to my two-sample t-test and report out the sample means then. But since we're doing notes, let's go ahead and go the longer route. Let's go to Stat → Calc → 1-Var Stats. I'm going to start with List 1. Just go in order. Enter, Enter, Enter. So my sample mean for Denver is 2.738. That's what I'm writing in my notes. Stat → Calc → 1-Var Stats. I'm going to do List 2. So 2nd → 2 → Enter, Enter, Enter. The Houston—it's 2.698. A little bit less than Denver. Stat → Calc → Enter → List 3. So 2nd → 3 → Enter. 2.9. All Right. Now let's go ahead and go do our tests. Stat → Tests. The two-sample t-test is right there—Option 4. I am going to go with data, right? We have our data, we have our sample means. We could have also not noted down the standard deviation, but not worth our time. We could just go ahead and use data. Yes. Let's start by comparing Denver to Houston—that's my List 1 versus List 2. So this looks good. Frequency lists look good. Null hypothesis looks good. It's never pooled. And I always like to go with draw. You can also see that my calculator is trying to draw something else. Let me fix that real quick. So Y = ... Yeah, my Plot 1 is turned on. Let me scroll up, hit Enter, turn that thing off. Go back: Stat → Tests → Option 4 → Draw. Okay. Now this looks better. So t = ... I'm going to note it down in my note... 2.7091. And my p-value is 0.0389. Let's do my other 2-Samp test. Option 4. All right. So I want to do Houston versus Cleveland. That's List 2 versus List 3. So 2nd → 2, 2nd → 3. Arrow down. Come over here to Draw. t-value is -5.0754. Oh, that is not matching—oh wait. No, that's okay. Five. Sorry, I had the solution key, but I realized I wrote my list in a slightly different order, which is why it wasn’t matching at first. Okay, that was good. [p-value is 0.0038.] Stat → Tests → Option 4. Make this the List 1 versus List 3. So Denver versus Cleveland. Note down that score and that p-value [t=-4.278, p=0.0116]. All right. Let me go back to my notes. I reported all of my sample means for Denver, Houston, and Cleveland in order. Looking at them, I can see that Houston is the smallest one. Cleveland is the largest. Right—when I compare Denver versus Houston, I do get a small p-value, but it's not small enough for my Bonferroni corrected significance level. So I'm going to fail to reject the null hypothesis. For these two though [Houston vs Cleveland and Denver vs Cleveland]—right—when we compare those p-values to this [the significance level], these ones are smaller. Right, which makes sense. Again, Cleveland is larger than these two. These were both the two smaller ones, and they're fairly close together, but there's a decent difference there between those sample means. And it turns out there's even a large enough difference here. Right—these two—not a very big difference. We failed to reject. Denver–Houston: not significantly different. But Cleveland versus Houston or Denver—it is significantly different. [write in notes if we reject or fail to reject the null hypothesis for each test.] Cleveland has a significantly higher average price than the other two cities. No significant difference though between Denver and Houston. So, if needed, we could do multiple two-sample tests. We just have to adjust our significance level. Right—otherwise we might say that there is a significant difference when maybe not that much of a difference. Do something similar if we were working with, say, confidence intervals. Right—we can repeat Example One by finding the Bonferroni-corrected intervals for all the comparisons.