Consider a large$-$sample level $0\mathrm{.}01$ test for testing $H_0$:$p=0\mathrm{.}2$ against $H_a$:$p>0\mathrm{.}2.$

$($a$)$ For the alternative value $p=0\mathrm{.}21,$ compute $(0\mathrm{.}21)$ for sample sizes $n=400,1600,12,100,40,000,$ and $90,000.($Round your answers to four decimal places.$)$

$($b$)$ For hat$(p)=\frac{x}{n}=0\mathrm{.}21,$ compute the $P-$value when $n=400,1600,12,100,$ and $40,000.($Round your answers to four decimal places.$)$

$($c$)$ In most situations, would it be reasonable to use a level $0\mathrm{.}01$ test in conjunction with a sample size of $40,000?$ Why or why not?

Yes, even when the departure from $H_0$ is significant from a practical point of view, a statistically significant result is not likely to appear; it is difficult for the test to detect departures from $H_0.$

Yes, it is always advantageous to have a very large sample size, because it will detect very small departures from $H_0.$

No$,$ even when the departure from $H_0$ is insignificant from a practical point of view, a statistically significant result is highly likely to appear; the test is too likely to detect small departures from $H_0.$

No$,$ it is never advantageous to have a very large sample size, because it cannot detect very small departures from $H_0.$