3 Ten randomly chosen athletes were coached for a 200 m event. For each athlete, the times taken to run 200 m before and after coaching were measured. The sample mean times before and after coaching were 23.43 seconds and 22.84 seconds respectively. For each athlete the difference, \(d\) seconds, in the times before and after coaching was calculated and an unbiased estimate of the population variance of \(d\) was found to be 0.548 . Stating any required assumption, test at the \(5 \%\) significance level whether the population mean time for the 200 m run decreased after coaching.

# Question 3:

| Answer/Working | Marks | Guidance |
|---|---|---|
| Assume population of differences has a normal distribution or sample random | B1 | Either assumption. AEF |
| $H_0: \mu_B - \mu_A = 0$, $H_1: \mu_B - \mu_A > 0$ | B1 | |
| $t = (23.43 - 22.84)/\sqrt{(0.548/10)}$ | M1 | |
| $= 2.520$ | | A1 | |
| $CV = 1.833$ | B1 | Seen |
| $2.52 >$ CV so reject $H_0$ | M1 | Allow from CV $2.262$ (2-tail) |
| $1.812, 1.734$ | | |
| Accept that there is evidence that mean time has reduced | A1$\sqrt{}$ | **7** | ft wrong CV |

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3 Ten randomly chosen athletes were coached for a 200 m event. For each athlete, the times taken to run 200 m before and after coaching were measured. The sample mean times before and after coaching were 23.43 seconds and 22.84 seconds respectively. For each athlete the difference, $d$ seconds, in the times before and after coaching was calculated and an unbiased estimate of the population variance of $d$ was found to be 0.548 . Stating any required assumption, test at the $5 \%$ significance level whether the population mean time for the 200 m run decreased after coaching.

\hfill \mbox{\textit{OCR S3 2006 Q3 [7]}}