Question

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?

Answer 1

Solution: The null and alternative hypotheses are:

$H_{0}:\sigma=0.004$

$H_{a}:\sigma<0.004$

Under null hypothesis, the test statistic is:

$\chi^{2}=\frac{(n-1)s^{2}}{\sigma^{2}}$

               $=\frac{(29-1)0.0036^{2}}{0.004^{2}}$

               $=22.68$

Now, the critical value of $\chi^{2}$ at 0.01 level of significance for $df=n-1=29-1=28$ is:

$\chi^{2}_{critical}=48.278$

Since the test statistic, $\chi^{2}=22.68$ is less than the critical value,$\chi^{2}_{critical}=48.278$, 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