The piston diameter of a certain hand pump is 0.8 inch. The manager determines that the diameters are normally distributed, with a mean of 0.8 inch and a standard deviation of 0.003 inch. After recalibrating the production machine, the manager randomly selects 24 pistons and determines that the standard deviation is 0.0022 inch. Is there significant evidence for the manager to conclude that the standard deviation has decreased at the $\alpha=0.05$ level of significance?
What are the correct hypotheses for this test?
The null hypothesis is $\mathrm{H}_{0}$ :
The alternative hypothesis is $\mathrm{H}_{1}$
Answer
The alternative hypothesis is \(\mathrm{H}_{1}: \sigma < 0.003\)
Step 1 ：We are testing whether the standard deviation has decreased. The null hypothesis is that the standard deviation has not decreased, i.e., it is still 0.003 inch. The alternative hypothesis is that the standard deviation has decreased, i.e., it is less than 0.003 inch.
Step 2 ：The null hypothesis is \(\mathrm{H}_{0}: \sigma = 0.003\)
Step 3 ：The alternative hypothesis is \(\mathrm{H}_{1}: \sigma < 0.003\)
