Construct a Confidence Interval For $\mu _ { \mathrm { d } }$

Construct a confidence interval for $\mu _ { \mathrm { d } }$ , the mean of the differences d for the population of paired data. Assume that the
population of paired differences is normally distributed.
-A coach uses a new technique in training middle distance runners. The times for 8 different athletes to run 800
meters before and after this training are shown below. $$\begin{array} { r | c c c c c c c c } \text { Athlete } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } & \text { G } & \text { H } \\\hline \text { Time before training (seconds) } & 112.7 & 115.2 & 109.2 & 117.2 & 113.9 & 118.7 & 115.2 & 110.4 \\\hline \text { Time after training (seconds) } & 113.3 & 113.9 & 106.8 & 118 & 112.1 & 118.8 & 111.6 & 106.5\end{array}$$ Using a 0.05 level of significance, test the claim that the training helps to improve the athletes' times for the 800
meters.