4 The time, $x$ seconds, spent by each of a random sample of 100 customers at an automatic teller machine (ATM) is recorded. The times are summarised in the table.

\begin{center}
\begin{tabular}{|l|l|}
\hline
Time (seconds) & Number of customers \\
\hline
$20 < x \leqslant 30$ & 2 \\
\hline
$30 < x \leqslant 40$ & 7 \\
\hline
$40 < x \leqslant 60$ & 18 \\
\hline
$60 < x \leqslant 80$ & 27 \\
\hline
$80 < x \leqslant 100$ & 23 \\
\hline
$100 < x \leqslant 120$ & 13 \\
\hline
$120 < x \leqslant 150$ & 7 \\
\hline
$150 < x \leqslant 180$ & 3 \\
\hline
Total & 100 \\
\hline
\end{tabular}
\end{center}

(a) Calculate estimates for the mean and standard deviation of the time spent at the ATM by a customer.\\
(b) The mean time spent at the ATM by a random sample of $\mathbf { 3 6 }$ customers is denoted by $\bar { Y }$.\\
(i) State why the distribution of $\bar { Y }$ is approximately normal.\\
(ii) Write down estimated values for the mean and standard error of $\bar { Y }$.\\
(iii) Hence estimate the probability that $\bar { Y }$ is less than $1 \frac { 1 } { 2 }$ minutes.

\hfill \mbox{\textit{AQA S1  Q4}}