The time needed for college students to complete a certain paper$-$and$-$pencil maze follows a normal distribution with a mean of $30$ seconds and a standard deviation of $2\mathrm{.}3$ seconds. You wish to see if the mean time $$ is changed by vigorous exercise, so you have a group of $20$ college students exercise vigorously for $30$ minutes and then complete the maze. It takes them an average of $\stackrel{}{x}=31$ seconds to complete the maze. Use this information to test the hypotheses

$H_0$:$=30$

$H_a$:$30$

Conduct a test using a significance level of $=0\mathrm{.}05.$

$($a$)$ The test statistic

$($b$)$ The positive critical value, $z^{***}=$

$($c$)$ The final conclusion is

A$.$ We can reject the null hypothesis that $=30$ and accept that $30.$

B$.$ There is not sufficient evidence to reject the null hypothesis that $=30.$