The table below shows the distribution of the waiting times for some customers in a...
The table below shows the distribution of the waiting times for some customers in a certain petrol station.
| Waiting time (in mins) | No of customers |
| 1.5 - 1.9 | 3 |
| 2.0 - 2.4 | 10 |
| 2.5 - 2.9 | 18 |
| 3.0 - 3.4 | 10 |
| 3.5 - 3.9 | 7 |
| 4.0 - 4.4 | 2 |
(a) Write down the class boundaries of the distribution.
(b) Construct a cumulative frequency curve for the data;
(c) Using your graph, estimate: (i) the interquartile range of the distribution ; (ii) the proportion of customers who could have waited for more than 3 minutes.
Explanation
| Waiting time (in mins) | Class boundaries | No of customers | Cum Freq |
| 1.5 - 1.9 | 1.45 - 1.95 | 3 | 3 |
| 2.0 - 2.4 | 1.95 - 2.45 | 10 | 13 |
| 2.5 - 2.9 | 2.45 - 2.95 | 18 | 31 |
| 3.0 - 3.4 | 2.95 - 3.45 | 10 | 41 |
| 3.5 - 3.9 | 3.45 - 3.95 | 7 | 48 |
| 4.0 - 4.4 | 3.95 - 4.45 | 2 | 50 |
(b)
(c) Interquartile range of the distribution = \(Q_{3} - Q_{1}\)
\(Q_{3} = \frac{3 \times 51}{4} = 38.25\)
\(Q_{1} = \frac{1 \times 51}{4} = 12.75\)
From the graph, \(Q_{3} = 3.25 ; Q_{1} = 2.45\)
\(\therefore \text{The interquartile range of the distribution } = 3.25 - 2.45 = 0.80\)
(ii) Proportion of customers who must have waited for more than 3 minutes = \(\frac{16}{50} = \frac{8}{25}\)