S1 June 2011 Q5
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 |
(You may use \(\sum \mathrm{f}x^2 = 8603.75\))
The teacher suggested that a normal distribution could be used to model the times taken by the students to complete the sudoku.
On another occasion the teacher calculated the quartiles for the times taken by the students to complete a different sudoku and found
\[Q_1 = 8.5 \qquad Q_2 = 13.0 \qquad Q_3 = 21.0\]| Scheme | Marks |
|---|---|
| 10.5 | B1 |
| (1) |
Notes
In parts (a) to (c) a correct answer with no working scores full marks for that value.
B1 for 10.5 which may be in the table
| Scheme | Marks |
|---|---|
| \((Q_2 =)\ \ (15.5 +)\ \dfrac{\frac{1}{2} \times 30 - 14}{8} \times 3\) or \(\dfrac{\frac{1}{2} \times 31 - 14}{8} \times 3\) | M1 |
| = 15.875 or 16.0625 | A1 |
| (2) |
Notes
M1 for a correct ratio and times 3, ignore the lower boundary for this mark
A1 for awrt 15.9 (if \(n\) =30 used) or awrt 16.1 (if \(n\)+1 = 31 is used)
| Scheme | Marks |
|---|---|
| \(\bar{x} = \dfrac{477.5}{30}\) = 15.9 \((15.91\dot{6})\) [ Accept \(\dfrac{191}{12}\) or \(15\tfrac{11}{12}\) ] | M1, A1 |
| \(\sigma = \sqrt{\dfrac{8603.75}{30} - \bar{x}^2}\) ,= 5.78 (accept \(s\) = 5.88) | M1A1ft, A1 |
| (5) |
Notes
1st M1 for attempt at \(\sum \mathrm{f}x\) (this may be seen in the table as f\(x\): 10, 73.5, 70, 136, 82, 106 [condone 1 slip] or awrt 500) and use of \(\dfrac{\sum \mathrm{f}x}{\sum \mathrm{f}}\) or a correct expression for mean.
1st A1 for awrt 15.9
2nd M1 for an attempt at \(\sigma\) or \(\sigma^2\), can ft their mean, condone mis-labelling \(\sigma^2 = \sqrt{\ldots}\) etc
Allow use of their \(\sum \mathrm{f}x^2\) (awrt 9000)
2nd A1ft for a correct expression including square root, ft their mean but not their \(\sum \mathrm{f}x^2\).
No label or correct label is OK but wrong label (e.g. \(\sigma^2 = \sqrt{\ldots}\) ) is A0
3rd A1 for awrt 5.78, allow \(s\) = awrt 5.88. SC Allow M1A1A0 for awrt 5.79 if \(\bar{x}\) correct
| Scheme | Marks |
|---|---|
| Since mean and median are similar (or equal or very close) a normal distribution may be suitable. [Allow mean or median close to mode/modal class] | B1 |
| (1) |
Notes
B1 for a reason implying or stating symmetry. "Time is continuous" or “evenly distributed” is B0
| Scheme | Marks |
|---|---|
| \(Q_3 - Q_2\,(= 8) \gt (4.5 =)\,Q_2 - Q_1\) | M1 |
| Therefore positive skew | A1 |
| (2) | |
| (11 marks) |
Notes
M1 for a clear reason or comparison, values not essential but comparison implying they have been found is required.
A1 for stating "positive skew". Condone just "positive" but "positive correlation" is A0
Do not allow arguments based on mean and median since this part relates to a different set of data.