4. The following table summarises the times, \(t\) minutes to the nearest minute, recorded for a group of students to complete an exam.
| Time (minutes) \(t\) | \(11 - 20\) | \(21 - 25\) | \(26 - 30\) | \(31 - 35\) | \(36 - 45\) | \(46 - 60\) |
| Number of students f | 62 | 88 | 16 | 13 | 11 | 10 |
$$\text { [You may use } \sum \mathrm { f } t ^ { 2 } = 134281.25 \text { ] }$$
- Estimate the mean and standard deviation of these data.
- Use linear interpolation to estimate the value of the median.
- Show that the estimated value of the lower quartile is 18.6 to 3 significant figures.
- Estimate the interquartile range of this distribution.
- Give a reason why the mean and standard deviation are not the most appropriate summary statistics to use with these data.
The person timing the exam made an error and each student actually took 5 minutes less than the times recorded above. The table below summarises the actual times.
| Time (minutes) \(t\) | \(6 - 15\) | \(16 - 20\) | \(21 - 25\) | \(26 - 30\) | \(31 - 40\) | \(41 - 55\) |
| Number of students f | 62 | 88 | 16 | 13 | 11 | 10 |
- Without further calculations, explain the effect this would have on each of the estimates found in parts (a), (b), (c) and (d).