Which test would you use to examine whether the mean difference in paired data differs from zero?

Which test would you use to examine whether the mean difference in paired data differs from zero?

• A. Matched pairs t-test
• B. 1-sample t-test
• C. 2-sample t-test
• D. 1-proportion z-test

Answer: (D)

To determine whether the average difference between paired observations differs from zero, focus on the differences within each pair. Compute D_i = X_i − Y_i for every pair, then test whether the mean of these differences is zero. The standard method is the matched pairs (paired) t-test. It uses the mean and variability of the differences across pairs, with the t-statistic t = (mean(D) − 0) / (sd(D)/sqrt(n)). This directly assesses the within-pair change and whether, on average, it is zero. Assumptions are that the differences are roughly normally distributed; with many pairs, the test remains reliable. If normality is questionable with a small sample, the Wilcoxon signed-rank test is a good nonparametric alternative. The other tests don’t fit this question as precisely. A two-sample t-test compares means of two independent groups and ignores pairing. A 1-proportion z-test focuses on a proportion rather than a mean. A simple 1-sample t-test on the raw data could test against a value, but it doesn’t directly address the mean difference within paired observations without first forming the differences. The paired approach is the right, direct way to test whether the mean difference is zero.

To determine whether the average difference between paired observations differs from zero, focus on the differences within each pair. Compute D_i = X_i − Y_i for every pair, then test whether the mean of these differences is zero. The standard method is the matched pairs (paired) t-test. It uses the mean and variability of the differences across pairs, with the t-statistic t = (mean(D) − 0) / (sd(D)/sqrt(n)). This directly assesses the within-pair change and whether, on average, it is zero. Assumptions are that the differences are roughly normally distributed; with many pairs, the test remains reliable. If normality is questionable with a small sample, the Wilcoxon signed-rank test is a good nonparametric alternative.

The other tests don’t fit this question as precisely. A two-sample t-test compares means of two independent groups and ignores pairing. A 1-proportion z-test focuses on a proportion rather than a mean. A simple 1-sample t-test on the raw data could test against a value, but it doesn’t directly address the mean difference within paired observations without first forming the differences. The paired approach is the right, direct way to test whether the mean difference is zero.