A marriage counselor has traditionally seen that the proportion $p$ of all married couples for whom her communication program can prevent divorce is $75\%.$ After making some recent changes, the marriage counselor now claims that her program can prevent divorce in more than $75\%$ of married couples. In a random sample of $225$ married couples who completed her program, $174$ of them stayed together. Based on this sample, is there enough evidence to support the marriage counselor's claim at the $0\mathrm{.}10$ level of significance?

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

Carry your intermediate computations to three or more decimal places. $($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$)$ Is there enough evidence to support the marriage counselor's claim that the proportion of married couples for whom her program can prevent divorce is more than $75\%?$

Yes No