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]
| Waiting time in minutes \(( x )\) | Frequency |
| \(0 - 3\) | 8 |
| \(3 - 5\) | 12 |
| \(5 - 6\) | 13 |
| \(6 - 8\) | 9 |
| \(8 - 12\) | 8 |
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{table}
- 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]
| Waiting Time | \(x < 3\) | \(3 - 5\) | \(5 - 6\) | \(6 - 8\) | \(x > 8\) |
| Expected Frequency | 8.56 | 12.73 | 7.56 | \(a\) | \(b\) |
\captionsetup{labelformat=empty}
\caption{Table 2}
- Find the value of \(a\) and the value of \(b\).
- Test, at the \(5 \%\) level of significance, the manager's belief. State your hypotheses clearly.