Answer:

1. (a) Using differences \(d=\text{before}-\text{after}\), the test statistic is \(t=2.39\) with \(8\) degrees of freedom. Since \(2.39\gt 1.860\), reject \(H_0\) at the \(5\%\) level. There is sufficient evidence to support the coach’s belief that the programme reduces race times.

2. (b) The paired \(t\)-test may not be appropriate because the population distribution of the paired differences may not be normal or symmetric, especially if the size of the improvement depends on how slow the athlete was initially.

3. (c) A suitable alternative would be a paired-sample sign test. A Wilcoxon matched-pairs signed-rank test could also be considered.

1. (a) Let \(d=\text{time before training}-\text{time after training}\), so that a positive value of \(d\) means the athlete’s time has reduced.

   The hypotheses are \(H_0:\mu_d=0\) and \(H_1:\mu_d\gt 0\), where \(\mu_d\) is the population mean reduction in time.

   The paired differences are

   \(5,\ 0,\ -1,\ 6,\ 1,\ -2,\ 8,\ 7,\ 15\).

   So \(n=9\), \(\sum d=39\), and \(\sum d^2=405\). Hence

   \(\bar d=\frac{39}{9}=\frac{13}{3}\).

   The unbiased sample variance of the differences is

   \(s_d^2=\frac{1}{9-1}\left(405-\frac{39^2}{9}\right)=\frac{1}{8}(405-169)=29.5\).

   Therefore

   \(s_d=\sqrt{29.5}\).

   The test statistic is

   \(t=\frac{\bar d-0}{s_d/\sqrt{n}}=\frac{13/3}{\sqrt{29.5}/3}=2.393\).

   For a one-tailed \(5\%\) test with \(9-1=8\) degrees of freedom, the critical value is \(1.860\). Since

   \(2.393\gt 1.860\),

   we reject \(H_0\). There is sufficient evidence at the \(5\%\) significance level to suggest that the new training programme reduces the athletes’ times, so the coach’s belief is supported.

2. (b) A paired \(t\)-test assumes that the population of paired differences is normally distributed. If slower athletes tend to improve by more than faster athletes, the differences may depend on the initial time and may not come from a normal or symmetric distribution. This makes the paired \(t\)-test less appropriate.

3. (c) A paired-sample sign test would be a suitable non-parametric alternative, since it uses only the direction of change within each pair. A Wilcoxon matched-pairs signed-rank test is another possible paired non-parametric test.