A data set includes data from student evaluations of courses. The summary statistics are $n=96, \bar{x}=3.58, s=0.63$. Use a 0.05 significance level to test the claim that the population of student course evaluations has a mean equal to 3.75 . Assume that a simple random sample has been selected. Identify the null and alternative hypotheses, test statistic, Pvalue, and state the final conclusion that addresses the original claim. What are the null and alternative hypotheses? A. \[ \begin{array}{l} H_{0}: \mu=3.75 \\ H_{1}: \mu<3.75 \end{array} \] C. \[ \begin{array}{l} H_{0}: \mu \neq 3.75 \\ H_{1}: \mu=3.75 \end{array} \] Determine the test statistic. -2.64 (Round to two decimal places as needed.) Determine the P-value. (Round to three decimal places as needed.) B. \[ \begin{array}{l} H_{0}: \mu=3.75 \\ H_{1}: \mu \neq 3.75 \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu=3.75 \\ H_{1}: \mu>3.75 \end{array} \]

Step 1 ：Identify the null and alternative hypotheses. The null hypothesis \(H_{0}\) is that the population mean \(\mu\) is equal to 3.75. The alternative hypothesis \(H_{1}\) is that the population mean \(\mu\) is not equal to 3.75. So, we have: \[ \begin{array}{l} H_{0}: \mu=3.75 \ H_{1}: \mu \neq 3.75 \end{array} \]

Step 2 ：Calculate the test statistic. The test statistic (z) is calculated using the formula \(z = \frac{\bar{x} - \mu_{0}}{s / \sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(\mu_{0}\) is the population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size. Substituting the given values, we get \(z = \frac{3.58 - 3.75}{0.63 / \sqrt{96}}\), which gives \(z \approx -2.64\).

Step 3 ：Calculate the P-value. The P-value is calculated using the cumulative distribution function (CDF) of the standard normal distribution. For a two-tailed test, we multiply the result by 2. The P-value is approximately 0.008.

Step 4 ：Compare the P-value with the significance level. The P-value (0.008) is less than the significance level (0.05), so we reject the null hypothesis.

Step 5 ：State the final conclusion. Since we rejected the null hypothesis, we conclude that the population mean is not equal to 3.75. Therefore, the final answer is: The test statistic is \(\boxed{-2.64}\) and the P-value is \(\boxed{0.008}\). We reject the null hypothesis and conclude that the population mean is not equal to 3.75.