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• Students need to learn how and when to use the chi-square tests. Just as before, there is value in understanding why the formula has its structure, but little value in simply memorizing the formula.
• However, students should be able to compute the expected counts in a goodness of fit test. (In other words, they should be able to perform some of the intermediate steps required to compute the statistic.) It's a very good learning excercise for students to do the same (compute expected counts) in the two-way table (for tests for independence or homogeneity).
• The null hypothesis in the test for independence is that the variables ARE independent. Rejecting the null means that you are deciding that they are NOT independent and therefore related. This strikes some students as a little backwards, because independence is made to seem so rare and interesting. But remind them that usually you want to prove that two variables are related (dependent), not unrelated (independent). So the null (rhymes with "dull") position would be that they are not related.
• The TI83/TI84 doesn't do goodness-of-fit tests automatically. But it can be coerced into doing so if you enter the data via two lists.
• It's tricky to teach students why the chi-square statistic in the context of a test of independence is a measure of lack of independence. If you want some more information on this, visit the NCSSM site and click on "Chi-square Analyses by Dan Teague, 2003".
• If you want to learn more about how to determine the degrees of freedom for chi-square tests, see Dan Teague's article mentioned in the previous bullet.
• The chi-square distribution with 1 degree of freedom is the distribution you would get if you took a standard normal random variable and squared it. Imagine this experiment: you want to see if eye color (dark/light) has a relationship to whether students wear corrective lenses. You can do a chi-square test of independence and you will get a chi-square statistic. You can also do a z-test to compare two proportions (using pooled proportions): is the proportion of light-eyed students with corrective lenses the same as the dark-eyed with corrective lenses? Take the z-statistic and square it -- you get the chi-squared statistic. Both statistics have the same p-values. You can show algebraically (should you wish) that these two test statistics are the same, but of course it only works for 2X2 tables. Want to see more?
• The "chi-square statistic" will only have (approximately) a chi-square distribution if the number of expected counts in all cells is at least 1, and no more than 20% of the expected counts are less than 5. That's a good rule-of-thumb to follow, similar to the np>10 and n*(1-p)>10 that we check before using the normal approximation to the distribution of p-hat. There are other rules of thumb published as well, and any of them will do. Students must use any of the published rules of thumb to check conditions on the AP exam.