The sports performance director at a university wishes to investigate whether there is a difference in the means of the specific gravities of blood of cyclists and runners. She models the distribution of specific gravity for cyclists as $N\left(\mu_c, 8^2\right)$ and for runners as $N\left(\mu_r, 10^2\right)$.

\begin{enumerate}[label=(\alph*)]
\item State suitable hypotheses for this investigation. [1]
\end{enumerate}

The mean specific gravity of blood of a random sample of 40 cyclists from the university was 1063. The mean specific gravity of blood of a random sample of 40 runners from the same university was 1060.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Calculate and interpret the $p$-value for the data. [6]

\item Suppose now that both samples were of size $n$, instead of 40. Find the least value of $n$ that would ensure that an observed difference of 3 in the mean specific gravities would be significant at the 1\% level. [4]
\end{enumerate}

\hfill \mbox{\textit{WJEC Further Unit 5 2024 Q4 [11]}}