Practice a College Board-style free response question on a paired (matched-pairs) t-test. Write your response, then reveal the model answer to see exactly what earns each point.
A coach records the resting heart rate (in beats per minute) of 10 randomly selected athletes both before and after a six-week training program. For each athlete she computes the difference (before − after). The 10 differences have a mean of d̄ = 4.2 bpm and a standard deviation of s_d = 3.1 bpm. A dotplot of the differences is roughly symmetric with no outliers. The coach wants to know whether the program reduces resting heart rate.
The appropriate procedure is a paired (matched-pairs) t-test, which is a one-sample t-test on the differences. The data are paired because each athlete is measured twice (before and after), so the two measurements are linked, not independent. Letting μ_d be the true mean difference (before − after), the hypotheses are H₀: μ_d = 0 versus Hₐ: μ_d > 0 (before is higher, meaning the program reduces heart rate).
Random: the athletes were randomly selected. 10%: 10 athletes are fewer than 10% of all athletes. Normal/Large Sample: the dotplot of differences is roughly symmetric with no outliers, so the t-procedure is appropriate. The test statistic is t = d̄/(s_d/√n) = 4.2/(3.1/√10) = 4.2/0.980 ≈ 4.28, with df = n − 1 = 9.
Because the p-value (0.001) is less than α = 0.05, we reject H₀. There is convincing evidence that the true mean difference in resting heart rate (before − after) is greater than 0 — that is, that the training program reduces athletes' resting heart rate on average. A Type I error here would mean concluding that the program reduces resting heart rate when in reality it has no effect (the true mean difference is 0).