As part of an investigation, a random sample was taken of 50 footballers who had completed an obstacle course in the early morning. The time taken by each of these footballers to complete the obstacle course, $x$ minutes, was recorded and the results are summarised by

$$\sum x = 1570 \quad \text{and} \quad \sum x^2 = 49467.58$$

\begin{enumerate}[label=(\alph*)]
\item Find unbiased estimates for the mean and variance of the time taken by footballers to complete the obstacle course in the early morning. [4]
\end{enumerate}

An independent random sample was taken of 50 footballers who had completed the same obstacle course in the late afternoon. The time taken by each of these footballers to complete the obstacle course, $y$ minutes, was recorded and the results are summarised as

$$\bar{y} = 30.9 \quad \text{and} \quad s_y^2 = 3.03$$

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Test, at the 5\% level of significance, whether or not the mean time taken by footballers to complete the obstacle course in the early morning, is greater than the mean time taken by footballers to complete the obstacle course in the late afternoon. State your hypotheses clearly. [7]
\item Explain the relevance of the Central Limit Theorem to the test in part (b). [1]
\item State an assumption you have made in carrying out the test in part (b). [1]
\end{enumerate}

\hfill \mbox{\textit{Edexcel S3 2015 Q6 [13]}}