A random sample of elementary school children in New York state is to be selected to estimate the proportion $p$ who have received a medical examination during the past year. An interval estimate of the proportion $p$ with a margin of error of $0.055$ and $95$% confidence is required.

(a) $\ $ Assuming no prior information about $\hat{p}$ is available, approximately how large of a sample size is needed?
$n =$

(b) $\ $ If a planning study indicates that $\hat{p}$ is around $0.8$, approximately how large of a sample size is needed?
$n =$