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to an initiative to recruit top students, an administrator at a college claims that this year's entering class must have a greater mean IQ score than that of entering classes from previous years. The administrator tests a random sample of $13$ of this year's entering students and finds that their mean IQ score is $115,$ with a standard deviation of $10.$ The college records indicate that the mean IQ score for entering students from previous years is $112.$

Is there enough evidence to conclude, at the $0\mathrm{.}10$ level of significance, that the population mean IQ score, $,$ of this year's class is greater than that of previous years? To answer, assume that the IQ scores of this year's entering class are approximately normally distributed.

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 $H_1.$

$H_0$:$=112$

$H_1$:$>112$

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

Degrees of freedom:

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

$0\mathrm{.}075$

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

$\backslash$table$[[,,p$