4. Customers at a post office are timed to see how long they wait until being served at the counter. A random sample of 50 customers is chosen and their waiting times, $x$ minutes, are summarised in Table 1.

\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | }
\hline
Waiting time in minutes $( x )$ & Frequency \\
\hline
$0 - 3$ & 8 \\
\hline
$3 - 5$ & 12 \\
\hline
$5 - 6$ & 13 \\
\hline
$6 - 8$ & 9 \\
\hline
$8 - 12$ & 8 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Show that an estimate of $\bar { x } = 5.49$ and an estimate of $s _ { x } ^ { 2 } = 6.88$

The post office manager believes that the customers' waiting times can be modelled by a normal distribution.\\
Assuming the data is normally distributed, she calculates the expected frequencies for these data and some of these frequencies are shown in Table 2.

\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
Waiting Time & $x < 3$ & $3 - 5$ & $5 - 6$ & $6 - 8$ & $x > 8$ \\
\hline
Expected Frequency & 8.56 & 12.73 & 7.56 & $a$ & $b$ \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 2}
\end{center}
\end{table}
\item Find the value of $a$ and the value of $b$.
\item Test, at the $5 \%$ level of significance, the manager's belief. State your hypotheses clearly.
\end{enumerate}

\hfill \mbox{\textit{Edexcel S3 2013 Q4 [14]}}