Nw $33.$ Determining Sample Size A physical therapist wants to determine the difference in the proportion of men and women who participate in regular, sustained physical activity. What sample size should be obtained if she wishes the estimate to be within $3$ percentage points with $95\%$ confidence, assuming that

$($a$)$ she uses the $1998$ estimates of $21\mathrm{.}9\%$ male and $19\mathrm{.}7\%$ female from the U$.$S$.$ National Center for Chronic Disease Prevention and Health Promotion?

$($b$)$ she does not use any prior estimates?

$34.$ Determining Sample Size An educator wants to determine the difference between the proportion of males and females who have completed four or more years of college. What sample size should be obtained if she wishes the estimate to be within $2$ percentage points with $90\%$ confidence, assuming that

$($a$)$ she uses the $1999$ estimates of $27\mathrm{.}5\%$ male and $23\mathrm{.}1\%$ female from the U$.$S$.$ Census Bureau?

$($b$)$ she does not use any prior estimates?

$35.$ Designing a Study Stock fund managers are investment professionals who decide which stocks should be part of a portfolio. In an article in the Wall Street Journal $("$Not a Stock$-$Picker's Market,"WSJ$,$ January $25,2014),$ the performance of stock fund managers was considered based on dispersion in the market. In the stock market, risk is measured by the standard deviation rate of return of stock $($dispersion$).$ When dispersion is low, then the rate of return of the stocks that make up the market are not as spread out. That is$,$ the return on Company $x$ is close to that of $Y$ is close to