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 0.004 inch. after recalibrating the production​ machine, the manager randomly selects 29 pistons and determines that the standard deviation is 0.0036 inch. is there significant evidence for the manager to conclude that the standard deviation has decreased at the alpha equals 0.01 level of​ significance?

Solution: The null and alternative hypotheses are:

Under null hypothesis, the test statistic is:

Now, the critical value of at 0.01 level of significance for is:

Since the test statistic, is less than the critical value,, therefore, we fail to reject the null hypothesis and there is not significant evidence for the manager to conclude that the standard deviation has decreased at the alpha equals 0.01

square root of 36/6 = 2.449489743

square root of 16/6 = 1.632993162

square root of 6 = 2.449489743

the answer is square root of 16/6 because the square root of 36/6 is the same answer as square root of 6 = 2.449489743