Answer: paired comparisons t-test.

## Explanation When testing the mean difference between two normally distributed populations using **dependent samples** (also called paired samples or matched pairs), and the population variances are unknown, the **paired comparisons t-test** is the most appropriate test. ### Key Points: 1. **Dependent Samples**: These are samples where each observation in one sample is paired with an observation in the other sample (e.g., before-and-after measurements on the same subjects, matched pairs, etc.). 2. **Unknown Population Variances**: When variances are unknown, we use t-tests rather than z-tests. 3. **Why Option B is Correct**: - The paired comparisons t-test (also called paired t-test or dependent samples t-test) is specifically designed for dependent samples. - It tests whether the mean difference between paired observations is significantly different from zero. - The test statistic is calculated as: $$t = \frac{\bar{d}}{s_d/\sqrt{n}}$$ where $\bar{d}$ is the mean of the differences, $s_d$ is the standard deviation of the differences, and $n$ is the number of pairs. 4. **Why Other Options are Incorrect**: - **Option A (chi-square test)**: Used for testing independence between categorical variables or goodness-of-fit tests, not for comparing means of continuous variables. - **Option C (t-test with pooled variance)**: This is appropriate for **independent samples** when we assume equal variances (two-sample t-test with pooled variance), not for dependent samples. ### When to Use Each Test: - **Independent samples with unknown but equal variances**: Two-sample t-test with pooled variance (Option C) - **Independent samples with unknown and unequal variances**: Two-sample t-test with separate variances (Welch's t-test) - **Dependent samples with unknown variances**: Paired comparisons t-test (Option B) Therefore, for dependent samples with unknown population variances, the paired comparisons t-test is the correct choice.

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