2. Listed below are the ages in years of 15 randomly selected race car drivers. \[ \begin{array}{lllllllllllllll} 32 & 32 & 33 & 33 & 41 & 29 & 38 & 32 & 33 & 23 & 27 & 45 & 52 & 29 & 25 \end{array} \] (a) Use a 0.025 level of significance to test the claim that the standard deviation of ages of all race car drivers is less than 12.5 years. $H_{0}$ : $H_{A}$ : Test Statistic: $p$-value: Conclusion:
Solution
Step 1 ：We are given the ages of 15 randomly selected race car drivers: 32, 32, 33, 33, 41, 29, 38, 32, 33, 23, 27, 45, 52, 29, 25.
Step 2 ：We are asked to test the claim that the standard deviation of ages of all race car drivers is less than 12.5 years at a 0.025 level of significance.
Step 3 ：We set up our null hypothesis \(H_{0}\) that the standard deviation of ages of all race car drivers is 12.5 years.
Step 4 ：Our alternative hypothesis \(H_{A}\) is that the standard deviation is less than 12.5 years.
Step 5 ：We calculate the sample standard deviation (s) of the given ages, which is approximately 7.67.
Step 6 ：We calculate the test statistic using the formula for the chi-square test statistic for standard deviation, which is \((n-1)s^2 / σ^2\), where n is the sample size, s is the sample standard deviation, and σ is the hypothesized population standard deviation. The test statistic is approximately 5.27.
Step 7 ：We calculate the p-value using the chi-square distribution with n-1 degrees of freedom. The p-value is approximately 0.98.
Step 8 ：Since the p-value is greater than the level of significance (0.025), we do not reject the null hypothesis.
Step 9 ：This means that we do not have enough evidence to support the claim that the standard deviation of ages of all race car drivers is less than 12.5 years.
Step 10 ：Final Answer: The test statistic is approximately \(\boxed{5.27}\) and the p-value is approximately \(\boxed{0.98}\). We do not reject the null hypothesis at the 0.025 level of significance. Therefore, we do not have enough evidence to support the claim that the standard deviation of ages of all race car drivers is less than 12.5 years.
