Question

The piston diameter of a certain hand pump is 0.4 inch. The manager determines that the diameters are normally distributed, with a mean of 0.4 inch and a standard deviation of 0.004 inch. After recalibrating the production machine, the manager randomly selects 21 pistons and determines that the standard deviation is 0.0035 inch. Is there significant evidence for the manager to conclude that the standard deviation has decreased at the α = 0.01 level of significance?

What are we looking for and what are we going to do with this?

Answer 1

The question aims to determine if the standard deviation of piston diameters has significantly decreased, using the Chi-Square test for variance and comparing the p-value to the level of significance (0.01).

Step-by-step explanation:

The question relates to a hypothesis test on variance. Specifically, we want to know if there is significant evidence to support the claim that the standard deviation of the piston diameters has decreased after recalibrating the production machine. Since we are comparing standard deviations, we would use the Chi-Square test for a variance. With the given sample standard deviation (0.0035 inch) and the sample size (21 pistons), we can compute the Chi-Square test statistic. Then we compare the test statistic to the Chi-Square distribution with n-1 degrees of freedom to find the p-value.

Given the level of significance (α = 0.01), we determine whether the p-value is lower than α. If it is, we reject the null hypothesis that the standard deviation has not changed, and conclude with evidence that the standard deviation has decreased. If the p-value is higher, we do not have enough evidence to say the standard deviation has decreased.