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270 Part IV: Building Strong Connections with Chi-Square Tests

Checking the conditions before you start

Every statistical technique seems to have a catch, and this case is no exception.
In order to use the Chi-square distribution to interpret your goodness-of-fit
statistic, you have to be sure you have enough information to work with in
each cell. The stats gurus usually recommend that the expected count for
each cell turns out to be greater than or equal to five. If it doesn’t, one option
is to combine categories to increase the numbers.
In the M&M’S example, the expected cell counts are all above seven (see
Table 15-3), so the conditions are met. If this weren’t the case, you should
have taken a larger sample size, because you calculate the expected cell
counts by taking the expected percentage in that cell times the sample size. If
you increase the sample size, you increase the expected cell count. A higher
sample size also increases your chances of detecting a real deviation from
the model. This idea is related to the power of the test (see Chapter 3 for
information on power).

After you collect your data, it’s not right to go back and take a new and larger
sample. It’s best to set up the appropriate sample size ahead of time, and you
can do this by determining what sample size you need to get the expected cell
counts to be at least five. For example, if you roll a fair die, you expect of the
outcomes to be ones. If you only take a sample of six rolls, you have an
expected cell count of , which isn’t enough. However, if you roll the die
30 times, your expected cell count is , which is just enough to meet
the condition.

The steps of the Chi-square

goodness-of-fit test

Assuming the necessary condition is met (see the previous section), you can
get down to actually conducting a formal goodness-of-fit test.

The general version of the null hypothesis for the goodness-of-fit test is Ho:
The model holds for all categories; versus the alternative hypothesis Ha: The
model doesn’t hold for at least one category. Each situation will dictate what
proportions should be listed in Ho for each category. For example, if you’re
rolling a fair die, you have Ho: Proportion of ones = ; proportion of
twos = ; . . . ; proportion of sixes = .

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