In the context of hypothesis testing, when is it appropriate to use the $t$-test for paired differences?

• A. When comparing the means of two independent samples with equal variances
• B. When analyzing the variance of more than two groups
• C. When comparing the means of two related samples, such as measurements taken before and after a treatment on the same subjects
• D. When testing the proportion of successes in a single sample

Answer: (C)

Verified step by step guidance

1. Understand the purpose of the paired difference t-test: it is used to compare the means of two related or dependent samples. This typically happens when the same subjects are measured twice under different conditions (e.g., before and after treatment).
2. Recognize that the paired difference t-test accounts for the fact that the two samples are not independent, but rather linked or paired, which reduces variability caused by individual differences.
3. Contrast this with other tests: for example, comparing means of two independent samples with equal variances requires an independent samples t-test, not a paired t-test.
4. Note that analyzing the variance of more than two groups is done using ANOVA (Analysis of Variance), which is different from the paired t-test.
5. Understand that testing the proportion of successes in a single sample involves tests for proportions, such as the one-sample z-test for proportions, and is unrelated to the paired difference t-test.