Before changes to its management staff, an automobile assembly line noperation had a scheduled mean completion time of $15\mathrm{.}2$ minutes. The standard deviation

of completion times was $1\mathrm{.}7$ minutes. An analyst at the company suspects that, under new management, the mean completion time, $,$ is now less than $15\mathrm{.}2$

minutes. To test this claim, a random sample of $70$ completion times under new management was taken by the analyst. The sample had a mean of $15$ minutes.

Can we support, at the $0\mathrm{.}10$ level of significance, the claim that the population mean completion time under new management is less than $15\mathrm{.}2$ minutes?

Assume that the population standard deviation of completion times has not changed under new management.

Perform a one$-$tailed test. Then complete the parts below.

Carry your intermediate computations to three or more decimal places, and round your responses as specified below. $($If necessary, consult a list of formulas.$)$

$($a$)$ State the null hypothesis $H_0$ and the alternative hypothesis $H_1.$

$H_0$:$$

$H_1$:$$

$($b$)$ Determine the type of test statistic to use.

$($c$)$ Find the value of the test statistic. $($Round to three or more decimal places.$)$

$($d$)$ Find the $p-$value. $($Round to three or more decimal places.$)$

$($e$)$ Can we support the claim that the population mean completion time under