As part of an experiment, a random sample of 60 runners recorded the time they took to complete a mountain trail in the summer. Their times, xxx minutes, are summarised by:
∑x=4680and∑x2=368 840 \sum x = 4680 \quad \text{and} \quad \sum x^2 = 368\,840 ∑x=4680and∑x2=368840Find unbiased estimates for the mean and variance of the time taken by runners in the summer.
An independent random sample was taken of 60 runners who completed the same trail in the winter. The completion times, yyy minutes, are summarised by:
yˉ=81.2andsy2=62.4 \bar{y} = 81.2 \quad \text{and} \quad s_y^2 = 62.4 yˉ=81.2andsy2=62.4Test, at the 5% level of significance, whether or not the mean time taken to complete the trail in the summer is different from the mean time taken in the winter. State your hypotheses clearly.
Explain the relevance of the Central Limit Theorem to the test in part (b).
State an assumption you have made about the population variances in carrying out the test in part (b).