Problem
Use a $\chi^{2}$-test to test the claim $\sigma<39$ at the $\alpha=0.05$ significance level using sample statistics $\mathrm{s}=36.3$ and $\mathrm{n}=12$. Assume the population is normally distributed. Identify the null and alternative hypotheses. A. \[ \begin{array}{l} H_{0}: \sigma \geq 39 \\ H_{a}: \sigma<39 \end{array} \] C. \[ \begin{array}{l} H_{0}: \sigma>39 \\ H_{a}: \sigma \leq 39 \end{array} \] B. \[ \begin{array}{l} H_{0}: \sigma<39 \\ H_{a}: \sigma \geq 39 \end{array} \] D. \[ \begin{array}{l} H_{0}: \sigma \leq 39 \\ H_{a}: \sigma>39 \end{array} \] Identify the standardized test statistic. 9.530 (Round to three decimal places as needed.) Identify the critical value(s). (Round to three decimal places as needed. Use a comma to separate answers as needed.)
Solution
Step 1 ：State the null and alternative hypotheses. The null hypothesis is \(H_{0}: \sigma \geq 39\) and the alternative hypothesis is \(H_{a}: \sigma<39\).
Step 2 ：Identify the standardized test statistic. The test statistic is a chi-square statistic, which is calculated using the sample standard deviation, the population standard deviation under the null hypothesis, and the sample size. The standardized test statistic is \(\boxed{9.53}\).
Step 3 ：Identify the critical value(s). The critical value is a point on the test statistic distribution that separates the region where the null hypothesis is not rejected from the region where the null hypothesis is rejected. The critical value is \(\boxed{4.575}\).
