In a derby, eight jockeys are riding a few exceptionally well-trained horses. As part of the race, the horses are timed as they run an obstacle course. The table shows the times (in seconds) of the horses in the first lap and the final lap of the race. At \(\alpha=0.05\), is there enough evidence to support the claim that the horses' running times have changed?

(a) identify the claim and state \(H_{0}\) and \(H_{a}\),

(b) find the critical value(s) and identify the rejection region(s),

(c) calculate \(\bar{d}\) and \(s_{d}\),

(d) find the standardized test statistic \(t\),

(e) decide whether to reject or fail to reject the null hypothesis,

(f) interpret the decision in the context of the original claim. 

Assume the samples are random and dependent, and the populations are normally distributed.

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Horse 1 2 3 4 5 6 7 8 Time 125.6 136.4 129.8 135.4 121.0 136.7 175.4 146.8 (first lap) Time 128.7 135.5 139.1 124.7 139.8 129.9 161.6 151.7 (final lap)