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Chi-Square
and F Distributions 10
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2. Chi-Square: Goodness of Fit
Section
10.2
Chi-Square: Goodness of Fit
Copyright © Cengage Learning. All rights reserved.

3. Focus Points
Set up a test to investigate how well a sample distribution fits a given distribution.
Use observed and expected frequencies to compute the sample χ2 statistic.
Find or estimate the P-value and complete the test.

4. Chi-Square: Goodness of Fit
Last year, the labor union bargaining agents listed five categories and asked each employee to mark the one most important to her or him. The categories and corresponding percentages of favorable responses are shown in Table 10-8.
Bargaining Categories (last year)
Table 10-8

5. Chi-Square: Goodness of Fit
The bargaining agents need to determine if the current distribution of responses “fits” last year’s distribution or if it is different.
In questions of this type, we are asking whether a population follows a specified distribution. In other words, we are testing the hypotheses

6. Chi-Square: Goodness of Fit
We use the chi-square distribution to test “goodness-of-fit” hypotheses. Just as with tests of independence, we compute the sample statistic:

7. Chi-Square: Goodness of Fit
Next we use the chi-square distribution table to estimate the P-value of the sample χ2 statistic. Finally, we compare the P-value to the level of significance  and conclude the test.
In the case of a goodness-of-fit test, we use the null hypothesis to compute the expected values for the categories.
Let’s look at the bargaining category problem to see how this is done.

8. Chi-Square: Goodness of Fit
In the bargaining category problem, the two hypotheses are
H0: The present distribution of responses is the same as last year’s.
H1: The present distribution of responses is different.
The null hypothesis tells us that the expected frequencies of the present response distribution should follow the percentages indicated in last year’s survey.
To test this hypothesis, a random sample of 500 employees was taken. If the null hypothesis is true, then there should be 4%, or 20 responses, out of the 500 rating vacation time as the most important bargaining issue.

9. Chi-Square: Goodness of Fit
Table 10-9 gives the other expected values and all the information necessary to compute the sample statistic χ2.
Observed and Expected Frequencies for Bargaining Categories
Table 10-9

10. Chi-Square: Goodness of Fit
We see that the sample statistic is
Larger values of the sample statistic χ2 indicate greater differences between the proposed distribution and the distribution followed by the sample.
The larger the χ2 statistic, the stronger the evidence to reject the null hypothesis that the population distribution fits the given distribution. Consequently, goodness-of-fit tests are always right-tailed tests.

11. Chi-Square: Goodness of Fit
To test the hypothesis that the present distribution of responses to bargaining categories is the same as last year’s, we use the chi-square distribution (Table 7 of Appendix II) to estimate the P-value of the sample statistic χ2 =
To estimate the P-value, we need to know the number of degrees of freedom.

12. Chi-Square: Goodness of Fit
In the case of a goodness-of-fit test, the degrees of freedom are found by the following formula.
Notice that when we compute the expected values E, we must use the null hypothesis to compute all but the last one. To compute the last one, we can subtract the previous expected values from the sample size.

13. Chi-Square: Goodness of Fit
For instance, for the bargaining issues, we could have found the number of responses for overtime policy by adding the other expected values and subtracting that sum from the sample size 500.
We would again get an expected value of 30 responses. The degrees of freedom, then, is the number of E values that must be computed by using the null hypothesis.
For the bargaining issues, we have
d.f. = 5 – 1 = 4
where k = 5 is the number of categories.

14. Chi-Square: Goodness of Fit
We now have the tools necessary Table 7 of Appendix II to estimate the P-value of χ2 = Figure 10-5 shows the P-value. In Table 7, we use the row headed by d.f. = 4.
We see that χ2 = falls between the entries and
P-value
Figure 10-5

15. Chi-Square: Goodness of Fit
Therefore, the P-value falls between the corresponding right-tail areas and Technology gives the P-value 
To test the hypothesis that the distribution of responses to bargaining issues is the same as last year’s at the 1% level of significance, we compare the P-value of the statistic to  = 0.01.

16. Chi-Square: Goodness of Fit
We see that the P-value is less than , so we reject the null hypothesis that the distribution of responses to bargaining issues is the same as last year’s.
Interpretation At the 1% level of significance, we can say that the evidence supports the conclusion that this year’s responses to the issues are different from last year’s.
Goodness-of-fit tests involve several steps that can be summarized as follows.

17. Chi-Square: Goodness of Fit
Procedure:

18. Chi-Square: Goodness of Fit
cont’d