EXAMPLE 9 Test for the homogeneity of proportions

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The American Academy of Pediatrics recommends that children's TV-watching time be limited to two hours or less per day. Here, we examine whether a relationship exists between watching TV for more than two hours per day and being overweight. The National Center for Health Statistics conducted a survey of children 12–15 years old. Three random samples were taken, one sample of normal or underweight children, one sample of overweight children, and one sample of obese children. The surveys noted whether the children watched TV more than two hours per day. The results are shown in Table 10.

Test whether the population proportions of children watching more than two hours per day of TV are the same for the three weight statuses, using the p-value method, Minitab, and level of significance .

tvandweight

Table 11.25: Table 10 Numbers watching more than two hours of TV, for three weight statuses
Normal or
underweight		 Overweight Obese Total
Number watching more
than two hours of TV		 140 44 82 266
Number watching
two hours or less of TV		 329 80 91 500
Total 469 124 173 766

Solution

The Minitab results are shown here in Figure 20.

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Figure 11.21: FIGURE 20 Minitab results for the test for homogeneity.

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We use the same steps as for the test for independence.

  • Step 1 State the hypotheses and the rejection rule. Check the conditions.

    • Not all the proportions in are equal.

    Reject if the p-value .

    The expected frequencies are shown in Figure 20. None of them are less than either 1 or 5. Therefore, the conditions are met, and we may proceed with the hypothesis test.

    Note: The conditions and the test statistic for the test for the homogeneity of proportions are the same as for the test for independence.

  • Step 2 Find the test statistic is shown as “Pearson Chi-Square = 17.207.” There are rows and columns, so the degrees of freedom are .
  • Step 3 Find the p-value. Minitab provides the p-value, which is essentially 0.000.
  • Step 4 State the conclusion and the interpretation. The p-value of 0.000 is less than . We therefore reject . Evidence exists, at level of significance , that not all population proportions of watching TV more than two hours per day are equal.

NOW YOU CAN DO

Exercises 19–26.