A researcher read that the average number of hours an adult sleeps on Friday night to Saturday morning was $7\mathrm{.}2$ hours on average. The researcher feels that college students do not sleep $7\mathrm{.}2$ hours on average. The researcher randomly selected $15$ students and found that on average they slept $8\mathrm{.}3$ hours. The standard deviation of the sample was $1\mathrm{.}2$ hours. At $=0\mathrm{.}05,$ is there enough evidence to say that college students do not sleep $7\mathrm{.}2$ hours on average? $(19$ points$)$

$($a$)$ State both hypotheses.

$($b$)$ Find the critical value$($s$).$

$($c$)$ Compute the test value.

$($d$)$ Make the decision to reject or not to reject the null hypothesis by stating "Reject" or "Not Reject".