Problem
Hypothesis test for the population mean: $t$ test using the p value method Thanks 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 21 of this year's entering students and finds that their mean IQ score is 112 , with a standard deviation of 12 . The college records indicate that the mean IQ score for entering students from previous years is 111 . Is there enough evidence to conclude, at the 0.10 level of significance, that the population mean IQ score, $\mu$, 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 $H_{1}$. $H_{0}: \square$ (b) Determine the type of test statistic to use. (choose one) $\nabla$ (d) Find the value of the test statistic. (Round to three or more decimal places.) (e) Can we conclude that the mean IQ score of this year's class is greater than that of previous years? Y Yes ONo \begin{tabular}{ccc} \hline$\mu$ & $\sigma$ & $p$ \\ $\bar{x}$ & $s$ & $\hat{p}$ \\ $\square$ & $\square \square$ & $\frac{\square}{\square}$ \\ $\square=\square$ & $\square \leq \square$ & $\square \geq \square$ \\ $\square \neq \square$ & $\square<\square$ & $\square>\square$ \\ $\times$ & & 0 \end{tabular}
Solution
Step 1 ：State the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$. The null hypothesis $H_{0}$: $\mu = 111$. The alternative hypothesis $H_{1}$: $\mu > 111$.
Step 2 ：Determine the type of test statistic to use. The type of test statistic to use is a t statistic.
Step 3 ：Find the value of the test statistic. The value of the test statistic is approximately $\boxed{0.382}$.
Step 4 ：Can we conclude that the mean IQ score of this year's class is greater than that of previous years? We cannot conclude that the mean IQ score of this year's class is greater than that of previous years. The p value is approximately $\boxed{0.353}$, which is greater than the significance level of 0.10. Therefore, we fail to reject the null hypothesis.
