4
\begin{enumerate}[label=(\alph*)]
\item Write down the number of ways of choosing 5 objects from 12 distinct objects.
\item Each possible set of 5 different integers selected from the integers $1,2 , \ldots , 12$ is obtained, and for each set, the sum of the 5 integers is found. The sum $S$ can take values between 15 and 50 inclusive. Part of the frequency distribution of $S$ is shown in the following table, together with the cumulative frequencies.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
S & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\
\hline
Frequency & 1 & 1 & 2 & 3 & 5 & 7 & 10 & 13 & 17 \\
\hline
Cumulative Frequency & 1 & 2 & 4 & 7 & 12 & 19 & 29 & 42 & 59 \\
\hline
\end{tabular}
\end{center}

Use these numbers to determine the critical region for a 1-tail Wilcoxon rank-sum test at the $2 \%$ significance level when $m = 5$ and $n = 7$.
\item A student says that, for a Wilcoxon rank-sum test on samples of size $m$ and $n$, where $m$ and $n$ are large, the mean and variance of the test statistic $R _ { m }$ are 200 and $616 \frac { 2 } { 3 }$ respectively.

Show that at least one of these values must be incorrect.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Statistics 2024 Q4 [6]}}