Suppose that you wish to estimate a population proportion and want to determine a sample size that will ensure a given margin of error for a $95 \%$ confidence interval. Suppose further that an educated guess of $0.3$ is used for $\hat { p }$ when determining the sample size. What values of the observed value of $\hat { p }$ will yield a larger margin of error than the one specified? A)$0.3

Question

Suppose that you wish to estimate a population proportion and want to determine a sample size that will ensure a given margin of error for a $95 \%$ confidence interval. Suppose further that an educated guess of $0.3$ is used for $\hat { p }$ when determining the sample size. What values of the observed value of $\hat { p }$ will yield a larger margin of error than the one specified?  
A)$0.3 < \hat { p } < 0.5$
B) $\hat { p } < 0.3$
C) $0.3 < \hat { p } < 0.7$
D) $\hat { p } > 0.3$

Quizplus · Accepted Answer

The margin of error is maximized when $\hat{p}(1-\hat{p})$ is maximized, which occurs at $\hat{p} = 0.5$. Thus, values of $\hat{p}$ between 0.3 and 0.7 will yield a larger margin of error than the one specified with $\hat{p} = 0.3$.

Suppose That You Wish to Estimate a Population Proportion and Want