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Chapter 15: Using Chi-Square Tests for Goodness-of-Fit 271

Following are the general steps for the Chi-square goodness-of-fit test, with
the M&M’S example illustrating how you can carry out each step:

1. Write down Ho using the percentages that you expect in your model
for each category.
Using a subscript to indicate the proportion (p) of M&M’S you expect
to fall into each category (see Table 15-1), your null hypothesis is Ho:
p = 0.13, p = 0.14, p = 0.13, p = 0.24, p = 0.20, and
brown yellow red blue orange
p = 0.16. All these proportions must hold in order for the model to
green
be upheld.
2. Write your Ha: This model doesn’t hold for at least one of the
percentages.
Your alternative hypothesis, Ha, in this case, would be: One (or more) of
the probabilities given in Ho isn’t correct. In other words, you conclude
that at least one of the colors of M&M’S has a different proportion than
what’s stated in the model.
3. Calculate the goodness-of-fit statistic using the steps in the earlier
section “Calculating the goodness-of-fit statistic.”
The goodness-of-fit statistic for M&M’S, from the earlier section, is 7.55.
As a reminder, you take the observed number in each cell minus the
expected number in that cell, square it, and divide by the expected
number in that cell. Do that for every cell in the table and add up
the results. For the M&M’S example, that total is equal to 7.55, the
goodness-of-fit statistic.
4. Look up the Chi-square distribution with k – 1 degrees of freedom,
where k is the number of categories you have.
You compare this statistic (7.55) to the Chi-square distribution with
6 – 1 = 5 degrees of freedom (because you have k = 6 possible colors of
M&M’S). (See Table A-3 in the appendix)
Looking at Figure 15-1 you can see that the value of 7.55 is nowhere
near the high end of this distribution, so you likely don’t have enough
evidence to reject the model provided by Mars for M&M’S colors.
5. Find the p-value of your goodness-of-fit statistic.
You use a Chi-square table to find the p-value of your test statistic
(see Table A-3 in the appendix). (For more info on the Chi-square
distribution, refer to Chapter 14.)
Because the Chi-square table can only list a certain number of results for
each of the degrees of freedom, the exact p-value for your test statistic
may fall between two p-values listed on the table.

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