1. **Problem Statement:** We have two independent samples of wait times from Clinic A and Clinic B. We need to test if their distributions differ using Anderson-Darling (AD) tests, check variance equality with Levene's test, and then choose and perform the appropriate hypothesis test. 2. **Step 1: Anderson-Darling (AD) Tests** - The AD test checks if a sample comes from a specified distribution, often normality. - Since both samples have size ≥ 25, the AD test is appropriate for checking normality. - Perform AD test on Clinic A and Clinic B samples separately to assess normality. 3. **Step 2: Levene's Test for Equality of Variances** - Levene's test checks if variances of two groups are equal. - Null hypothesis $H_0$: variances are equal. - Alternative hypothesis $H_a$: variances are not equal. - If Levene's test p-value < significance level (e.g., 0.05), reject $H_0$ and conclude variances differ. 4. **Step 3: Choosing the Test and Stating Hypotheses** - If both samples are approximately normal and variances are equal, use the two-sample t-test assuming equal variances. - If normality fails or variances differ, use the Welch's t-test or a nonparametric test like Mann-Whitney U. - Hypotheses for mean difference test: $$H_0: \mu_A = \mu_B$$ $$H_a: \mu_A \neq \mu_B$$ 5. **Step 4: Compute Test Statistic and p-value** - For two-sample t-test (equal variances): $$t = \frac{\bar{x}_A - \bar{x}_B}{s_p \sqrt{\frac{1}{n_A} + \frac{1}{n_B}}}$$ where $$s_p = \sqrt{\frac{(n_A - 1)s_A^2 + (n_B - 1)s_B^2}{n_A + n_B - 2}}$$ - For Welch's t-test (unequal variances): $$t = \frac{\bar{x}_A - \bar{x}_B}{\sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}}$$ - Calculate degrees of freedom accordingly. - Compute p-value from t-distribution. 6. **Step 5: Final Conclusion** - Based on p-value and significance level (e.g., 0.05), decide whether to reject $H_0$. - Discuss if variance equality materially affects test choice and interpretation. **Note:** Actual numeric calculations require data processing in software (e.g., R or Python). This explanation guides the approach and interpretation.