$(25\%)$ The new tax structure is supposed to help small businesses survive. It is known that

presently $20\%$ of all small businesses fail in their first year. In a random sample of $n=200$ new

small businesses followed for one year after opening, $30$ were recorded as filed.

$($a$)$ If we wish to test the claim that the new tax laws help small businesses survive how would we

express $H_0$ and $H_a?$

$($b$)$ Find the value of the test statistic based on hat$(p),$ the sample proportion of failed small businesses.

$($c$)$ What is the rejection region based on a significance level of $=0\mathrm{.}05?$ Can you conclude that

the new tax structure helps small businesses survive?

$($d$)$ Find the $p-$value for the test. Does your conclusion in part c change?

$($e$)$ For the considered test, compute a one$-$sided confidence bound and draw your conclusion.