To complete the hypothesis testing for the given null and alternative hypotheses, we will follow these steps:

Given:

• Null Hypothesis $H_0: \sigma^2 = 100$
• Alternative Hypothesis $H_A: \sigma^2 \neq 100$
• Sample size $n = 24$
• Sample standard deviation $s = 99$
• Significance level $\alpha = 0.10$

Part a: Decision Rule

1. Determine the Degrees of Freedom (df): $df = n - 1 = 24 - 1 = 23$

2. Find the Critical Values: For a two-tailed test with $\alpha = 0.10$:

   • The critical value at $\alpha/2 = 0.05$ for the upper tail and lower tail.

   • Look up the chi-square distribution table or use a calculator for $\chi^2_{0.05, 23}$ and $\chi^2_{0.95, 23}$.

   • Critical Values:

     • $\chi^2_{0.05, 23} \approx 36.415$ (upper critical value)
     • $\chi^2_{0.95, 23} \approx 10.982$ (lower critical value)

3. State the Decision Rule:

   • Correct choice: $B. \text{If the test statistic } \chi^2 > 36.415, \text{ reject } H_0 \text{ or if the test statistic } \chi^2 < 10.982, \text{ reject } H_0.$

Summary of Decision Rule

• Reject $H_0$ if $\chi^2 > 36.415$ or $\chi^2 < 10.982$.

Next Steps

Would you like to calculate the test statistic using the given sample standard deviation, or do you have any specific questions about the hypothesis test? Here are some questions that might help clarify your understanding:

1. What is the formula to calculate the test statistic for this hypothesis test?
2. How do you interpret the critical values in context of the hypothesis test?
3. What does it mean to reject or fail to reject the null hypothesis?
4. How would the decision rule change if the significance level were different?
5. What are the assumptions of the chi-square test for variance?

Tip: Always check the assumptions of the test before performing hypothesis testing to ensure the results are valid.