Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. In addition, in each case for which the results are slgnificant, state which group (1 or 2 ) has the larger mean. Hypotheses: \( H_{0}: \mu_{1}=\mu_{2} v s H_{a}: \mu_{1} \neq \mu_{2} \) (a) \( 95 \% \) confidence interval for \( \mu_{1}-\mu_{2}: 0.16 \) to 0.54 Conclusion: Gignificance level: Group with the larger mean: \( 99 \% \) confidence interval for \( \mu_{1}-\mu_{2}:-2.5 \) to 5.2 (b) 9.2 Significance level: Group with the larger mean: \( \square \) (c) \( 90 \% \) confidence interval for \( \mu_{1}-\mu_{2}:-9.6 \) to -3.2 Conclusion: Significance level: \( \square \)

To analyze the given confidence intervals and draw conclusions about the hypotheses, we will follow these steps: 1. **Identify the hypotheses**: - Null hypothesis (\( H_{0} \)): \( \mu_{1} = \mu_{2} \) - Alternative hypothesis (\( H_{a} \)): \( \mu_{1} \neq \mu_{2} \) 2. **Interpret the confidence intervals**: - If the confidence interval does not include 0, we reject the null hypothesis. - If the confidence interval includes 0, we fail to reject the null hypothesis. 3. **Determine the significance level**: - The significance level corresponds to the confidence level used. For example, a \( 95\% \) confidence interval corresponds to a significance level of \( \alpha = 0.05 \). 4. **Identify which group has the larger mean**: - If the entire confidence interval is above 0, then group 1 has a larger mean. - If the entire confidence interval is below 0, then group 2 has a larger mean. Now, let's analyze each case: ### (a) \( 95\% \) confidence interval for \( \mu_{1} - \mu_{2}: 0.16 \) to 0.54 - **Conclusion**: Since the interval \( (0.16, 0.54) \) does not include 0, we reject the null hypothesis. - **Significance level**: \( 0.05 \) (corresponding to \( 95\% \) confidence). - **Group with the larger mean**: Group 1 has the larger mean. ### (b) \( 99\% \) confidence interval for \( \mu_{1} - \mu_{2}: -2.5 \) to 5.2 - **Conclusion**: Since the interval \( (-2.5, 5.2) \) includes 0, we fail to reject the null hypothesis. - **Significance level**: \( 0.01 \) (corresponding to \( 99\% \) confidence). - **Group with the larger mean**: Not applicable, as we fail to reject the null hypothesis. ### (c) \( 90\% \) confidence interval for \( \mu_{1} - \mu_{2}: -9.6 \) to -3.2 - **Conclusion**: Since the interval \( (-9.6, -3.2) \) does not include 0, we reject the null hypothesis. - **Significance level**: \( 0.10 \) (corresponding to \( 90\% \) confidence). - **Group with the larger mean**: Group 2 has the larger mean. ### Summary of Results: - **(a)**: - Conclusion: Reject \( H_{0} \) - Significance level: \( 0.05 \) - Group with the larger mean: Group 1 - **(b)**: - Conclusion: Fail to reject \( H_{0} \) - Significance level: \( 0.01 \) - Group with the larger mean: Not applicable - **(c)**: - Conclusion: Reject \( H_{0} \) - Significance level: \( 0.10 \) - Group with the larger mean: Group 2