$7\mathrm{.}5\mathrm{.}17.$ A stock analyst claims to have devised a mathemat$-$

ical technique for selecting high$-$quality mutual funds and

promises that a client's portfolio will have higher average

ten$-$year annualized returns and lower volatility; that is$,$

a smaller standard deviation. After ten years, one of the

analyst's twenty$-$four$-$stock portfolios showed an average

ten$-$year annualized return of $11\mathrm{.}50\%$ and a standard de$-$

viation of $10\mathrm{.}17\%.$ The benchmarks for the type of funds

considered are a mean of $10\mathrm{.}10\%$ and a standard deviation

of $15\mathrm{.}67\%.$

$($a$)$ Let $$ be the mean for a twenty$-$four$-$stock portfolio se$-$

lected by the analyst's method. Test at the $0\mathrm{.}05$ level that

the portfolio beat the benchmark; that is$,$ test $H_0$:$=10\mathrm{.}1$

versus $H_1$:$>10\mathrm{.}1.$

$($b$)$ Let $$ be the standard deviation for a twenty$-$four$-$

stock portfolio selected by the analyst's method. Test at

the $0\mathrm{.}05$ level that the portfolio beat the benchmark; that

is$,$ test $H_0$:$=15\mathrm{.}67$ versus $H_1$: