The piston diameter of a certain hand pump is 0.7 inch. The manager determines that the diameters are normally distributed, with a mean of 0.7 inch and a standard deviation of 0.006 inch. After recalibrating the production machine, the manager randomly selects 28 pistons and determines that the standard deviation is 0.0054 inch. Is there significant evidence for the manager to conclude that the standard deviation has decreased at the $\alpha=0.01$ level of significance?
Calculate the value of the test statistic.
$\chi^{2}=($ Round to three decimal places as needed.)
Use technology to determine the P-value for the test statistic.

Step 1 ：We are given that the sample size n = 28, the sample standard deviation s = 0.0054, and the hypothesized population standard deviation sigma = 0.006.

Step 2 ：The null hypothesis is that the standard deviation has not changed, i.e., \(\sigma = 0.006\), and the alternative hypothesis is that the standard deviation has decreased, i.e., \(\sigma < 0.006\).

Step 3 ：The test statistic for this hypothesis test is a chi-square statistic, which is calculated as \(\chi^{2} = (n-1)s^{2}/\sigma^{2}\).

Step 4 ：Substituting the given values into the formula, we get \(\chi^{2} = (28-1)(0.0054)^{2}/(0.006)^{2} = 21.87\).

Step 5 ：We can use a chi-square distribution with n-1 degrees of freedom to find the p-value. The p-value is 0.744.

Step 6 ：Since the p-value is greater than the significance level of 0.01, we do not reject the null hypothesis.

Step 7 ：There is not enough evidence to conclude that the standard deviation has decreased.

Step 8 ：Final Answer: The test statistic is \(\boxed{21.87}\) and the p-value is \(\boxed{0.744}\).