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?
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?
Since the P-value is
than the level of significance,
the null hypothesis. There
sufficient evidence to conclude that the
standard deviation has decreased at the 0.10 level of significance.

Step 1 ：The problem is asking to test the hypothesis that the standard deviation of the piston diameters has decreased after recalibration. The null hypothesis is that the standard deviation has not decreased, and the alternative hypothesis is that it has decreased.

Step 2 ：We can use a chi-square test to test this hypothesis. The test statistic for a chi-square test is calculated as \((n-1)s^2 / σ^2\), where n is the sample size, s is the sample standard deviation, and σ is the population standard deviation.

Step 3 ：We can then use the chi-square distribution to find the p-value associated with this test statistic. If the p-value is less than the significance level, we reject the null hypothesis and conclude that the standard deviation has decreased.

Step 4 ：Given that n = 21, s = 0.0033, and sigma = 0.005, we can calculate the test statistic as \((21-1) * 0.0033^2 / 0.005^2\), which gives us a test statistic of approximately 8.712.

Step 5 ：We can then use the chi-square distribution to find the p-value associated with this test statistic, which is approximately 0.986.

Step 6 ：Since the p-value is greater than the significance level of 0.10, we fail to reject the null hypothesis. This means that there is not sufficient evidence to conclude that the standard deviation has decreased at the 0.10 level of significance.

Step 7 ：Final Answer: The test statistic is \(\boxed{8.712}\) (rounded to three decimal places) and the p-value is \(\boxed{0.986}\) (rounded to three decimal places). Since the p-value is greater than the level of significance, we fail to reject the null hypothesis. There is not sufficient evidence to conclude that the standard deviation has decreased at the 0.10 level of significance.