The piston diameter of a certain hand pump is 0.6 inch. The manager determines that the diameters are normally distributed, with a mean of 0.6 inch and a standard deviation of 0.005 inch. After recalibrating the production machine, the manager randomly selects 21 pistons and determines that the standard deviation is 0.0033 inch. Is there significant evidence for the manager to conclude that the standard deviation has decreased at the $\alpha=0.10$ level of significance? What are the correct hypotheses for this test? The null hypothesis is $\mathrm{H}_{0}$ : The alternative hypothesis is $\mathrm{H}_{1}$ : Calculate the value of the test statistic. $\chi^{2}=\square$ (Round to three decimal places as needed.) Use technology to determine the P-value for the test statistic. The P-value is (Round to three decimal places as needed.) What is the correct conclusion at the $\alpha=0.10$ level of significance?
Solution
Step 1 ：The null hypothesis is \(H_{0} : \sigma = 0.005\).
Step 2 ：The alternative hypothesis is \(H_{1} : \sigma < 0.005\).
Step 3 ：Calculate the value of the test statistic using the formula \(\chi^{2} = \frac{(n-1)s^{2}}{\sigma^{2}}\), where n is the sample size, s is the sample standard deviation, and \(\sigma\) is the population standard deviation. In this case, n = 21, s = 0.0033, and \(\sigma\) = 0.005.
Step 4 ：The value of the test statistic is \(\chi^{2} = \boxed{8.712}\).
Step 5 ：Use technology to determine the P-value for the test statistic. The P-value is \(\boxed{0.986}\).
Step 6 ：Compare the P-value to the significance level \(\alpha\). Since the P-value is greater than the significance level of 0.10, we fail to reject the null hypothesis.
Step 7 ：There is not enough evidence to conclude that the standard deviation has decreased at the \(\alpha=0.10\) level of significance.
