A math teacher claims that she has developed a review course that increases the scores of students on the math portion of a college entrance exam. Based on data from the administrator of the exam, scores are normally distributed with $\mu=516$. The teacher obtains a random sample of 2000 students, puts them through the review class, and finds that the mean math score of the 2000 students is 521 with a standard deviation of 116 . Complete parts (a) through (d) below. (a) State the null and alternative hypotheses \[ \mathrm{H}_{0}: \mu=516, \mathrm{H}_{1}: \mu \mathrm{H} \rightarrow \mathbf{5 1 6} \] (b) Test the hypothesis at the $\alpha=0.10$ level of significance. Is a mean math score of 521 statistically significantly higher than 516 ? Conduct a hypothesis test using the P-value approach. Find the test statistic: \[ t_{0}=\square \] (Round to two decimal places as needed) an example Get more help Clear all Check answer

Step 1 ：State the null and alternative hypotheses: \(\mathrm{H}_{0}: \mu=516, \mathrm{H}_{1}: \mu > 516\)

Step 2 ：Calculate the test statistic using the formula for a one-sample t-test: \(t_{0} = \frac{x_{\text{bar}} - \mu}{s / \sqrt{n}}\)

Step 3 ：Substitute the given values into the formula: \(t_{0} = \frac{521 - 516}{116 / \sqrt{2000}} \approx 1.93\)

Step 4 ：Calculate the p-value using the t-distribution and the degrees of freedom (df = n - 1 = 1999): p-value \approx 0.027

Step 5 ：Compare the p-value to the level of significance: 0.027 < 0.10

Step 6 ：Since the p-value is less than the level of significance, reject the null hypothesis

Step 7 ：Conclude that the mean score after the review course is significantly higher than the original mean score

Step 8 ：Final Answer: The test statistic is approximately \(\boxed{1.93}\) and the p-value is approximately \(\boxed{0.027}\). Since the p-value is less than the level of significance (0.10), we reject the null hypothesis and conclude that the mean score after the review course is significantly higher than the original mean score.