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.
| Time (seconds) | Number of customers |
| \(20 < x \leqslant 30\) | 2 |
| \(30 < x \leqslant 40\) | 7 |
| \(40 < x \leqslant 60\) | 18 |
| \(60 < x \leqslant 80\) | 27 |
| \(80 < x \leqslant 100\) | 23 |
| \(100 < x \leqslant 120\) | 13 |
| \(120 < x \leqslant 150\) | 7 |
| \(150 < x \leqslant 180\) | 3 |
| Total | 100 |
- Calculate estimates for the mean and standard deviation of the time spent at the ATM by a customer.
- The mean time spent at the ATM by a random sample of \(\mathbf { 3 6 }\) customers is denoted by \(\bar { Y }\).
- State why the distribution of \(\bar { Y }\) is approximately normal.
- Write down estimated values for the mean and standard error of \(\bar { Y }\).
- Hence estimate the probability that \(\bar { Y }\) is less than \(1 \frac { 1 } { 2 }\) minutes.