Suppose that you can make reasonably good educated guesses, p1g and p2g, for the observed values of

Suppose that you can make reasonably good educated guesses, pˆ1g and pˆ2g, for the observed values of pˆ1 and pˆ2.

a. Use your result from Exercise 12.132 to show that a (1 − α)-level confidence interval for the difference between two population proportions that has an approximate margin of error of E can be obtained by choosing n1 = n2 =

pˆ1g(1 − pˆ1g ) + pˆ2g (1 − pˆ2g )

 zα/2 E

2 rounded up to the nearest whole number. Note: If you know likely ranges instead of exact educated guesses for the observed values of the two sample proportions, use the values in the ranges closest to 0.5 as the educated guesses.

b. Explain why the formula in part

(a) yields smaller (or at worst the same) sample sizes than the formula in Exercise 12.133.

c. When reasonably good educated guesses for the observed values of pˆ1 and pˆ2 can be made, explain why choosing the sample sizes by using the formula in part

(a) is preferable to choosing them by using the formula in Exercise 12.133.