For Exercises 19–22, test whether or not the proportions of successes are the same for all populations.

  1. State the hypotheses.
  2. Calculate the expected frequencies and verify that the conditions for performing the test for homogeneity of proportions are met.
  3. Find and state the rejection rule. Use level of significance .
  4. Find .
  5. Compare with . State the conclusion and the interpretation.

Question 11.56

21.

Sample 1 Sample 2 Sample 3 Sample 4
Successes 10 15 20 25
Failures 15 24 32 40

11.2.21

(a) . : Not all the proportions in are equal.

(b)

Sample 1 Sample 2 Sample 3 Sample 4 Total
Successes 9.67 15.08 20.11 25.14 70
Failures 15.33 23.92 31.89 39.86 111
Total 25 39 52 65 181

Since none of the expected frequencies is less than 1 and none of the expected frequencies is less than 5, the conditions for performing the test for homogeneity of proportions are met. (c) 7.815. Reject if . (d) 0.0215 (e) Since is not , we do not reject . There is insufficient evidence that not all the proportions in are equal.