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 0.006 inch. After recalibrating the production machine, the manager randomly selects 28 pistons and determines that the standard deviation is 0.0052 incl 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 $\mathrm{H}_{0}: \quad \sigma=0.006$. The alternative hypothesis is $\mathrm{H}_{1}: \quad \sigma<0.006$. Calculate the value of the test statistic. $2=$ (Round to three decimal places as needed.)

Step 1 ：The null hypothesis is \(\mathrm{H}_{0}: \sigma=0.006\).

Step 2 ：The alternative hypothesis is \(\mathrm{H}_{1}: \sigma<0.006\).

Step 3 ：The test statistic for a hypothesis test about a population standard deviation or variance is a chi-square statistic. The formula for the test statistic is \(\chi^2 = \frac{(n - 1)s^2}{\sigma^2}\), where n is the sample size, s is the sample standard deviation, and \(\sigma\) is the population standard deviation.

Step 4 ：In this case, n = 28, s = 0.0052, and \(\sigma\) = 0.006. We can substitute these values into the formula to calculate the test statistic.

Step 5 ：\(\chi^2 = \frac{(28 - 1) * 0.0052^2}{0.006^2} = 20.28\)

Step 6 ：The calculated chi-square statistic is approximately 20.28. This is the value of the test statistic that we will compare to the critical value to determine whether or not to reject the null hypothesis.

Step 7 ：Final Answer: The value of the test statistic is approximately \(\boxed{20.28}\).