Question 1

When should I use a paired t-test?

Accepted Answer

Use a paired t-test when: (1) the same subjects are measured under two conditions (e.g., blood pressure before and after medication), (2) subjects are matched in pairs based on relevant characteristics (e.g., twins given different treatments), or (3) repeated measurements are taken on the same unit (e.g., left vs right eye vision). The paired design eliminates between-subject variability, giving more statistical power than a two-sample independent t-test.

Question 2

What is the difference between a paired t-test and a two-sample t-test?

Accepted Answer

The key difference is the study design. In a paired t-test, each observation in Group 1 is linked to a specific observation in Group 2 (same subject or matched pair). The test works on the difference scores d_i = x_i − y_i. In a two-sample (independent) t-test, the two groups are unrelated. Using a two-sample test on paired data throws away information about the pairing and reduces statistical power.

Question 3

What is the null hypothesis of a paired t-test?

Accepted Answer

H₀: μ_d = 0 — the mean difference in the population is zero (no treatment effect). H₁: μ_d ≠ 0 for a two-tailed test, or H₁: μ_d > 0 / μ_d < 0 for one-tailed tests. Rejecting H₀ means there is evidence of a real difference between the two conditions.

Question 4

What are the assumptions of a paired t-test?

Accepted Answer

1) The difference scores d_i = x_i − y_i are independently sampled. 2) The differences are approximately normally distributed. This assumption can be relaxed with n ≥ 30 due to CLT. 3) The pairs are randomly selected. The test is fairly robust to non-normality of the original variables — it only requires the differences to be roughly normal.

Question 5

How do I interpret Cohen's d for a paired t-test?

Accepted Answer

Cohen's d = d̄ / s_d measures how many standard deviation units the mean difference is away from zero. Thresholds: d = 0.2 small, d = 0.5 medium, d = 0.8 large. A large d means the treatment had a sizeable effect relative to the natural variability in differences. Always report d alongside p-values — a small p with d = 0.1 signals a significant but practically tiny effect.

Question 6

What is the 95% confidence interval for the mean difference?

Accepted Answer

The 95% CI for μ_d is: d̄ ± t*(s_d / √n), where t* is the critical t-value for df = n−1 at α = 0.05. If the CI excludes zero, H₀ is rejected at α = 0.05. The CI tells you the plausible range for the true mean treatment effect in the population.

Scenario	Use	Example
Same subjects before/after	Paired t-test	Blood pressure before and after medication
Matched pairs by characteristic	Paired t-test	Twins assigned to different diets
Same unit measured twice	Paired t-test	Left vs right eye visual acuity
Two independent, unrelated groups	Two-sample t-test	Male vs female heights
Differences are non-normal	Wilcoxon signed-rank	Pain ratings (1–10 scale)

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