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The Goodness of Fit Test

I have found that students often have an easier time with chi-squared tests than with many of the other tests. Also, the chi-squared distribution can be intuitively constructed from binomial trials. For those reasons, I choose to introduce the chi-squared distribution before the t or the F distributions. The other nice thing about working with the chi-squared test is that the experiments are quick and easy to conduct. After presenting the information I will have a chart up at the front of the class everyday polling the students. Students are now asked to formulate the null hypothesis (which I have already developed to choose the poll question) and then perform the chi-squared test and make a conclusion.

A bit of caution about something in the text. Students are going to want to hold on to a particular chi-squared value to apply to all situations, regardless of the degrees of freedom or level of confidence. The book doesn’t help as it almost sounds like it declares that a particular chi-squared value is the threshold for rejection. I always make sure that students have to work different problems so they don’t develop a habit.  

Test of Independence

It is debatable whether you should keep this topic separate from the previous, or if they should be merged. The purpose of merging the two topics is that they are really the same. The difference in the test for independence is the same as fit, except for the way the null hypothesis is written. A reason to give it its own treatment is because the AP examination treats it more in that manner. I tend to treat them as one section. Conceptually, the difference is in how the null hypothesis is written and correlation is shown the same way that independence is. Also, while other sections can be helped by being stretched out a little, students do not need as much practice with chi-squared tests as many of the others.

A fun activity I remember from university is that we ran chi-squared tests for homogeneity on various random number generators. Even with different seeds, the random generator on the calculator did not fare well in our experiment. I can be a very fun one for students, especially if human responses are added in.

Testing one Variance

The text mentions that the F-test is sensitive to non-normality, which is only partially true. In this chapter it is, as the specific instance of the F-test being sensitive is when showing that variances are the same. The F-test is actually guarded pretty well against non-normality in other methods of testing, which is why it is a part of ANOVA, which is specifically mentioned as a robust test. I don’t know how much I would bother students with these details, but since a good student is diligent about checking requisite conditions, it may be worth mentioning.

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