5. A class of students had a sudoku competition. The time taken for each student to complete the sudoku was recorded to the nearest minute and the results are summarised in the table below.
| Time | Mid-point, \(x\) | Frequency, f |
| 2-8 | 5 | 2 |
| 9-12 | | 7 |
| 13-15 | 14 | 5 |
| 16-18 | 17 | 8 |
| 19-22 | 20.5 | 4 |
| 23-30 | 26.5 | 4 |
$$\text { (You may use } \sum \mathrm { f } x ^ { 2 } = 8603.75 \text { ) }$$
- Write down the mid-point for the 9-12 interval.
- Use linear interpolation to estimate the median time taken by the students.
- Estimate the mean and standard deviation of the times taken by the students.
The teacher suggested that a normal distribution could be used to model the times taken by the students to complete the sudoku.
- Give a reason to support the use of a normal distribution in this case.
On another occasion the teacher calculated the quartiles for the times taken by the students to complete a different sudoku and found
$$Q _ { 1 } = 8.5 \quad Q _ { 2 } = 13.0 \quad Q _ { 3 } = 21.0$$
- Describe, giving a reason, the skewness of the times on this occasion.