Before changes to its management staff, an automobile assembly line operation had a scheduled mean completion time of 12.6 minutes. The standard deviation of completion times was 1.8 minutes. An analyst at the company suspects that, under new management, the mean completion time, $\mu$, is now less than 12.6 Espanol) minutes. To test this claim, a random sample of 15 completion times under new management was taken by the analyst. The sample had a mean of 12.1 minutes. Assume that the population is normally distributed. Can we support, at the 0.10 level of significance, the claim that the population mean completion time under new management is less than 12.6 minutes? Assume that the population standard deviation of completion times has not changed under new management. Perform a one-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places, and round your responses as specified below. (If necessary, consult a list of formulas.) (a) State the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$. (b) Determine the type of test statistic to use. (c) Find the value of the test statistic. (Round to three or more decimal places.) (d) Find the p-value. (Round to three or more decimal places.) (e) Can we support the claim that the population mean completion time under new management is less than 12.6 minutes? OYes ONo \begin{tabular}{ccc} $\mu$ & $\sigma$ & $p$ \\ $\bar{x}$ & $s$ & $\hat{p}$ \\ $\square$ & $\square_{\square}$ & $\frac{\square}{\square}$ \\ $\square=\square$ & $\square \leq \square$ & $\square \geq \square$ \\ $\square \neq \square$ & $\square<\square$ & $\square>\square$ \\ $\times$ & & $S$ \end{tabular} Check Save For Later Submit Assignment
Solution
Step 1 ：The null hypothesis $H_{0}$ is that the population mean under the new management is equal to 12.6 minutes. The alternative hypothesis $H_{1}$ is that the population mean under the new management is less than 12.6 minutes.
Step 2 ：The type of test statistic to use is a z-test.
Step 3 ：To calculate the test statistic, we need to subtract the population mean from the sample mean, and then divide by the standard error of the mean. The standard error of the mean is the standard deviation divided by the square root of the sample size. The value of the test statistic is approximately -1.076.
Step 4 ：After calculating the test statistic, we can find the p-value using a z-table or a statistical software. The p-value is approximately 0.141.
Step 5 ：Since the p-value is greater than the significance level of 0.10, we fail to reject the null hypothesis. This means that we do not have enough evidence to support the claim that the population mean completion time under new management is less than 12.6 minutes.
Step 6 ：We cannot support the claim that the population mean completion time under new management is less than 12.6 minutes. Therefore, the final answer is \(\boxed{No}\).
