5 Jane, who supplies fruit to a jam manufacturer, knows that the weight of fruit in boxes that she sends to the manufacturer can be modelled by a normal distribution with unknown mean, $\mu$ grams, and unknown standard deviation, $\sigma$ grams.

Jane selects a random sample of 16 boxes and, using the $t$-distribution, calculates correctly that a $98 \%$ confidence interval for $\mu$ is $( 70.65,80.35 )$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that the sample mean is 75.5 grams.
\item Find the width of the confidence interval.
\item Calculate an estimate of the standard error of the mean.
\item Hence, or otherwise, show that an unbiased estimate of $\sigma ^ { 2 }$ is 55.6 , correct to three significant figures.
\end{enumerate}\item Jane decides that the width of the $98 \%$ confidence interval is too large.

Construct a $95 \%$ confidence interval for $\mu$, based on her sample of 16 boxes.
\item Jane is informed that the manufacturer would prefer the confidence interval to have a width of at most 5 grams.
\begin{enumerate}[label=(\roman*)]
\item Write down a confidence interval for $\mu$, again based on Jane's sample of 16 boxes, which has a width of 5 grams.
\item Determine the percentage confidence level for your interval in part (c)(i).
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{AQA S2 2009 Q5 [13]}}