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.