The piston diameter of a certain hand pump is 0.5 inch. The manager determines that the diameters are normally distributed, with a mean of 0.5 inch and a standard deviation of 0.007 inch. After recalibrating the production machine, the manager randomly selects 29 pistons and determines that the standard deviation is 0.0063 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 $\mathrm{H}_{0}: \sigma=0.007$. The alternative hypothesis is $\mathrm{H}_{1}: \sigma \quad \sigma[<0.007$. 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.)

Step 1 ：State the hypotheses. The null hypothesis is \(H_{0}: \sigma=0.007\). The alternative hypothesis is \(H_{1}: \sigma < 0.007\).

Step 2 ：Calculate the test statistic using the formula \(\chi^{2} = (n-1)\left(\frac{s}{\sigma}\right)^{2}\), where \(n\) is the sample size, \(s\) is the sample standard deviation, and \(\sigma\) is the population standard deviation. Substituting the given values, we get \(\chi^{2} = (29-1)\left(\frac{0.0063}{0.007}\right)^{2} = 22.68\).

Step 3 ：Use a chi-square distribution table or statistical software to find the p-value corresponding to the test statistic. The p-value is approximately 0.749.

Step 4 ：Compare the p-value with the significance level \(\alpha\). If the p-value is less than \(\alpha\), reject the null hypothesis. If the p-value is greater than \(\alpha\), fail to reject the null hypothesis. In this case, the p-value (0.749) is greater than the significance level (0.10), so we fail to reject the null hypothesis.

Step 5 ：Interpret the result. Since we failed to reject the null hypothesis, there is not enough evidence to conclude that the standard deviation has decreased. Therefore, the final answer is: The test statistic is \(\boxed{22.68}\) and the p-value is \(\boxed{0.749}\). 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 enough evidence to conclude that the standard deviation has decreased.