11-15. If the population proportion p is to be estimated by a 100(1 )% confidence interval...

11-15. If the population proportion p is to be estimated by a 100(1 − α)% confidence interval and we are sampling from a binomial population of size N without replacement, then a measure of how precisely we have estimated p is determined by ± error bound, where the error bound is w

2

= zα/2

p(1 − p)

n

N − n N − 1

.

Solving this expression for n yields n = mN m + N − 1

, where m = zα/2 p(1 − p)

(w/2)2 What is the interpretation of n? (Note: since p is unknown, use ˆp; if no prior or pilot estimate such as ˆp is available, then use p = 1 2 .) Suppose that a union shop involving N = 2500 members is interested in assessing union support for a new contract involving enhanced benefits rather than wage increases. To estimate the proportion p in favor of the contract, how large a sample is required so that the 95% confidence interval for p will not exceed 5%?