7 Mr Alott and Miss Fewer work in a postal sorting office.
\begin{enumerate}[label=(\alph*)]
\item The number of letters per batch, $R$, sorted incorrectly by Mr Alott when sorting batches of 50 letters may be modelled by the distribution $\mathrm { B } ( 50,0.15 )$.

Determine:
\begin{enumerate}[label=(\roman*)]
\item $\mathrm { P } ( R < 10 )$;
\item $\mathrm { P } ( 5 \leqslant R \leqslant 10 )$.
\end{enumerate}\item It is assumed that the probability that Miss Fewer sorts a letter incorrectly is 0.06 , and that her sorting of a letter incorrectly is independent from letter to letter.
\begin{enumerate}[label=(\roman*)]
\item Calculate the probability that, when sorting a batch of $\mathbf { 2 2 }$ letters, Miss Fewer sorts exactly 2 letters incorrectly.
\item Calculate the probability that, when sorting a batch of $\mathbf { 3 5 }$ letters, Miss Fewer sorts at least 1 letter incorrectly.
\item Calculate the mean and the variance for the number of letters sorted correctly by Miss Fewer when she sorts a batch of $\mathbf { 1 2 0 }$ letters.
\item Miss Fewer sorts a random sample of 20 batches, each containing 120 letters. The number of letters sorted correctly per batch has a mean of 112.8 and a variance of 56.86 .

Comment on the assumptions that the probability that Miss Fewer sorts a letter incorrectly is 0.06 , and that her sorting of a letter incorrectly is independent from letter to letter.

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\includegraphics[max width=\textwidth, alt={}]{adf1c0d2-b0a6-4a2f-baf2-cfb45d771315-15_2484_1709_223_153}
\end{center}
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA S1 2009 Q7 [14]}}