Did the 8-week program improve 5 km times?

Question

A coach times the same 12 runners on a 5 km course before and after an 8-week program. For each runner, compute the improvement (before time minus after time, in seconds). The improvements have a mean of 22 s and a sample standard deviation of 30 s.

Run a paired t-test at the 5% level (two-sided): did the program change times? Report the t-statistic.

Accepted Answer

Because each runner is measured twice, the two columns are paired. Reduce to the per-runner differences di and run a one-sample t-test on them, with n - 1 = 11 degrees of freedom. Hypotheses: H0: \mud = 0 (no change) versus H1: \mud 
eq 0. Standard error of the mean difference: 	ext{SE} = \frac{sd}{\sqrt{n}} = \frac{30}{\sqrt{12}} = \frac{30}{3.464} = 8.660 	ext{ s}. Test statistic: t = \frac{\bar d}{	ext{SE}} = \frac{22}{8.660} = 2.54. The two-sided 5% critical value for 11 degrees of freedom is t{0.025,\,11} = 2.201. Since 2.54 2.201, reject H0. The p-value is about 0.027. Conclusion: t…

Did the 8-week program improve 5 km times?

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