A college entrance exam company determined that a score of 20 on the mathematics portion of the exam suggests that a student is ready for college-level mathematics. To achieve this goal, the company recommends that students take a core curriculum of math courses in high school. Suppose a random sample of 200 students who completed this core set of courses results in a mean math score of 20.2 on the college entrance exam with a standard deviation of 3.6. Do these results suggest that students who complete the core curriculum are ready for college-level mathematics? That is, are they scoring above 20 on the mathematics portion of the exam? Complete parts a) through d) below. Click the icon to view the table of critical t-values. (a) State the appropriate null and altemative hypotheses. Fill in the correct answers below. The appropriate null and alternative hypotheses are $\mathrm{H}_{0}$ : $\mu=20$ versus $\mathrm{H}_{1}$ : $\mu>20$. (Type integers or decimals. Do not round.) (b) Verify that the requirements to perform the test using the t-distribution are satisfied. Select all that apply. A. The sample size is larger than 30 . B. The students' test scores were independent of one another. C. A boxplot of the sample data shows no outliers. D. The sample data confe from a population that is approximately normal. E. The students were randomly sampled. F. None of the requirements are satisfied. (c) Use the P-value approach at the $\alpha=0.10$ level of significance to test the hypotheses in part (a). Identify the test statistic. \[ t_{0}= \] (Round to two decimal places as needed.)

Step 1 ：State the appropriate null and alternative hypotheses. The appropriate null and alternative hypotheses are \(H_{0} : \mu=20\) versus \(H_{1} : \mu>20\).

Step 2 ：Verify that the requirements to perform the test using the t-distribution are satisfied. The sample size is larger than 30, the students' test scores were independent of one another, a boxplot of the sample data shows no outliers, the sample data come from a population that is approximately normal, and the students were randomly sampled.

Step 3 ：Identify the test statistic. The test statistic for a one-sample t-test is calculated as (sample mean - population mean) / (sample standard deviation / sqrt(sample size)). The sample mean is 20.2, the population mean is 20, the sample standard deviation is 3.6, and the sample size is 200. The test statistic is approximately \(0.79\).

Step 4 ：Use the P-value approach at the \(\alpha=0.10\) level of significance to test the hypotheses. The p-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. If the p-value is less than the significance level (0.10 in this case), we reject the null hypothesis. The p-value is approximately 0.216. This is greater than the significance level of 0.10, so we do not reject the null hypothesis.

Step 5 ：Interpret the results. This suggests that the mean score of students who completed the core curriculum is not significantly greater than 20. Therefore, the results do not provide strong evidence to suggest that students who complete the core curriculum are ready for college-level mathematics, at least not at the 0.10 significance level.

Step 6 ：Final Answer: The test statistic is approximately \(\boxed{0.79}\) and the p-value is approximately \(\boxed{0.216}\). Therefore, we do not reject the null hypothesis that the mean score is 20. The results do not provide strong evidence to suggest that students who complete the core curriculum are ready for college-level mathematics, at least not at the 0.10 significance level.