Suppose the mean IQ score of people in a certain country is 101 . Suppose the director of a college obtains a simple random sample of 39 students from that country and finds the mean IQ is 104.3 with a standard deviation of 13.8 Complete parts (a) through (d) below. (a) Consider the hypotheses $\mathrm{H}_{0}: \mu=101$ versus $\mathrm{H}_{1}, \mu>101$. Explain what the director is testing. Perform the test at the $\alpha=0.05$ level of significance. Write a conclusion for the test Explain what the director is testing. Choose the correct answer below. A. The director is testing if the sample provided sufficient evidence that the population mean IQ score is actually not greater than 101 B. The director is testing if the sample provided sufficient evidence that the population mean IQ score is actually equal to 101 c. The director is testing if the sample provided sufficient evidence that the population mean IQ score is actually greater than 101. D. The director is testing if the sample provided sufficient evidence that the population mean IQ score is actually not equal to 101
Solution
Step 1 ：The director is testing the hypothesis that the population mean IQ score is greater than 101. This is a one-tailed test where the null hypothesis is that the population mean is 101 and the alternative hypothesis is that the population mean is greater than 101.
Step 2 ：The director will use the sample mean, sample standard deviation, and sample size to calculate the test statistic and compare it to the critical value at the 0.05 level of significance to determine whether to reject or fail to reject the null hypothesis.
Step 3 ：Given that the population mean (mu) is 101, the sample mean (x_bar) is 104.3, the sample standard deviation (s) is 13.8, and the sample size (n) is 39.
Step 4 ：The test statistic (t) is calculated to be approximately 1.493, and the critical value (t_critical) at the 0.05 level of significance is approximately 1.686.
Step 5 ：Since the test statistic (t) is less than the critical value (t_critical), we fail to reject the null hypothesis.
Step 6 ：This means that we do not have sufficient evidence to conclude that the population mean IQ score is greater than 101.
Step 7 ：\(\boxed{\text{The director is testing if the sample provided sufficient evidence that the population mean IQ score is actually not greater than 101}}\)
