We are conducting a hypothesis test to determine if the standard deviation ($\sigma$) of the diameters of the golf balls is less than 0.004 inches, as required by the professional golfing association. Since the population is assumed to be normally distributed, this is a one-tailed test for the population standard deviation.

The hypotheses for this test are as follows:

• Null Hypothesis $H_0$: $\sigma = 0.004$ (The standard deviation is equal to 0.004 inches, implying no evidence that the standard deviation is less than the requirement.)
• Alternative Hypothesis $H_1$: $\sigma < 0.004$ (The standard deviation is less than 0.004 inches, meeting the requirement.)

Notation:

• $H_0$: $\sigma = 0.004$
• $H_1$: $\sigma < 0.004$

This is a left-tailed hypothesis test because we are testing if the standard deviation is less than the specified value.

Do you want to proceed with the calculation of the test statistic and decision rule for the hypothesis test?

Additional Questions:

1. How do we calculate the sample standard deviation?
2. What distribution should be used to test hypotheses about population variance?
3. How do we determine the critical value at $\alpha = 0.10$?
4. What is the difference between a one-tailed and two-tailed test?
5. How does sample size affect the test for standard deviation?

Tip:

When testing for variance or standard deviation, always use the chi-squared distribution if the population is assumed to be normally distributed.