8.

A company has a customer services call centre. The company believes that the time taken to complete a call to the call centre may be modelled by a normal distribution with mean 16 minutes and standard deviation $\sigma$ minutes.

Given that $10 \%$ of the calls take longer than 22 minutes,
\begin{enumerate}[label=(\alph*)]
\item show that, to 3 significant figures, the value of $\sigma$ is 4.68
\item Calculate the percentage of calls that take less than 13 minutes.

A supervisor in the call centre claims that the mean call time is less than 16 minutes. He collects data on his own call times.

\begin{itemize}
  \item $20 \%$ of the supervisor's calls take more than 17 minutes to complete.
  \item $10 \%$ of the supervisor's calls take less than 8 minutes to complete.
\end{itemize}

Assuming that the time the supervisor takes to complete a call may be modelled by a normal distribution,
\item estimate the mean and the standard deviation of the time taken by the supervisor to complete a call.
\item State, giving a reason, whether or not the calculations in part (c) support the supervisor's claim.

\section*{9.}
A fast food company has a scratchcard competition. It has ordered scratchcards for the competition and requested that $45 \%$ of the scratchcards be winning scratchcards.

A random sample of 20 of the scratchcards is collected from each of 8 of the fast food company's stores.

Assuming that $45 \%$ of the scratchcards are winning scratchcards, calculate the probability that in at least 2 of the 8 stores, 12 or more of the scratchcards are winning scratchcards.\\[0pt]
[Total 5 marks]

END OF TEST
\end{enumerate}

\hfill \mbox{\textit{SPS SPS FM Statistics 2020 Q8 [11]}}