Example one: A six-sided die was tossed 180 times, and the data is shown in the table below. Test the hypothesis the die is not fair using a significance level of 0.05. Okay, we've really already seen this data. We've done this calculation, so we can go a little bit faster. n = 180. Safe assumption is that the die is fair. That's what we'll use for our expected count. So, null hypothesis: the die is fair. The observed equals expected. The alternative hypothesis is that the die is not fair. The null hypothesis is what we'll use. If the die is fair, then we know the probability of any one side equals 1/6. And if we have 180, 1/6 of 180 is 30. Prepare: What is alpha? Well, we know alpha is 0.05. That was given to us. Random sample? Yes. Oh, you're not actually going to have to show that for me. Independent measurements? Yeah. Large sample? Yes. All of our expected values are 30, which is greater than 5. That’s really the only one you're going to need to show for me. Compute: We're going to go ahead and do our Chi-square goodness of fit test. Once we do our test, we'll spit out a chi-square value and a p. All right, let me go ahead and show my graphing calculator. All right, let's clear all this stuff out. Start fresh. So, Stat → Tests. Here's a bunch of hypothesis tests—it's actually not all of them. To get to my other, like this goodness of fit test, I find it better to scroll up. You could scroll down if you wanted to, but up is a little bit faster. And there we can see I have option D: the chi-square goodness of fit test. Again, this is a TI-84, so I have that feature. TI-83 generally won't have this, and so we'll have to do this calculation differently. We already had our observed and expected data in there from before. If you get to this point and realize, "Oh wait, I haven't typed in my observed and expected," just come over here to second mode → Stat → Enter and enter your data. So, like I said, we already had the observed data in list one, the expected data in list two. If not, go ahead and do it again now. Then go to Stat → Tests → Goodness of Fit Test. I like to scroll up from the bottom. So, my observed data is actually in list one. I'm going to type in here second and the number 1 to get list one. My expected data is in list two, so second 2 to get list two. Degrees of freedom: I haven't actually calculated that. So off on my notes, I'm going to write DF = number of categories - 1. Our number of categories here are the sides of our die. There are how many sides? Six. So, 6 - 1 would give me a degrees of freedom of 5. There we go. Feel free to use calculator Draw. I will always prefer Draw because I do—I am a visual person. I like to see that. I like to see that that chi-square value is where they're starting the shading. Like if I count over: 1, 2, 3, 4, 5, 6, 7, 8 and a half is where the shading has started, and here's the probability of that chi-square value. So I put that in my notes: chi-square = 8.5, p-value = 0.1307. If I had gone with the Calculate option, I would have more decimals, but it's the one I prefer. Let's go back to my notes. All right, so ignoring the actual test and the test statistic, to interpret we just look at that p-value, and we look at our significance level, as we always have. So looking at these, answer for yourself: do we reject or fail to reject? Surprisingly, if you think back to what we did in the 10.1 notes, we felt like this was a pretty large chi-square value. In this case, we do end up failing to reject, right? Our p-value is larger than our significance level, so we fail to reject. Because in this case, this is a large chi-square value for a degree of freedom of, say, 1, but for a degree of freedom of 5, it's actually not that large. Because we fail to reject, our conclusion is: we don't have enough evidence. We think the die might not be fair—that's our alternative hypothesis. We definitely see that these are not exactly the same numbers, but the difference isn't enough for us to really be worried, right? Even though there were a lot of twos, that could happen after 180 times. If you really don't think this die is fair, we need a larger sample. It turns out 180 is just not enough in this case. Now that you've seen a chi-square goodness of fit test, let's try another.