June 2025 Paper 3 Q2
2. Runners in an athletics club can train with either coach \(A\) or coach \(B\) for the 400 m race.
Coach \(A\) trains 120 runners for the 400 m and records the best time, \(x\) seconds, for each runner.
The results are summarised by the following statistics
\[\sum x = 6612 \qquad \sum x^2 = 364\,902\]The mean and standard deviation for the best times of the 100 runners trained for the 400 m by coach \(B\) are 55.1 seconds and 3.6 seconds respectively.
A 400 m race consists of equal numbers of the fastest runners trained by coach \(A\) and by coach \(B\)
| Scheme | Marks | AO |
|---|---|---|
| \(\left[\bar{x} = \dfrac{6612}{120}\right] = \mathbf{55.1}\) (seconds) | B1 | 1.1b |
| (1) |
Notes
Ignore units in this question
B1: for 55.1 or \(55\frac{1}{10}\) only (may be seen in the question) ISW e.g. 55.1 followed by 55 etc
Do not accept \(\frac{551}{10}\) that scores B0
| Scheme | Marks | AO |
|---|---|---|
| \(\left[\sigma_x =\right] \sqrt{\dfrac{364\,902}{120} - (\text{``}55.1\text{''})^2}\) or \(\sqrt{4.84}\) | M1 | 1.1b |
| \(= \mathbf{2.2}\) (seconds) | A1 | 1.1b |
| (2) |
Notes
Ignore units in this question
M1: for a correct expression (values seen not just a formula)(including \(\sqrt{\phantom{x}}\))
A1: for 2.2 or \(\frac{11}{5}\) o.e. (allow \(s = 2.209\ldots\) so awrt 2.21)
Correct answer is 2 marks
| Scheme | Marks | AO |
|---|---|---|
| Coach \(B\) because standard deviation is greater | M1 | 2.4 |
| This means \(B\) has a greater range and therefore more extreme values | A1ft | 2.2b |
| (2) | ||
| (5 marks) |
Notes
Ignore units in this question
Ignore comments about different number of runners. Don’t need to state means are the same.
If there is no answer given for (b) then no marks can be scored in (c)
“2.2” < 3.6
B1: for coach \(B\) and states s.d. of \(B\) is greater o.e. Must compare. Allow implied choose \(B\)
e.g. \(B\) has a greater range or \(B\)s values are more spread out or \(3.6 > \text{``}2.2\text{''}\) or \(\sigma_B > \text{``}2.2\text{''}\)
May say s.d. of \(A\) is smaller which is equivalent to saying s.d. of \(B\) is larger
dB1: (dep on 1st B1 scored) for clear reason why greater s.d. means choose coach \(B\)
Idea of greater range and more extreme values (faster times) in \(B\)
e.g. \(B\) has more (or greater chance of or greater % of) runners at lower times
(may compare mean – 1 or 2 or 3 standard deviations to show that \(B\) has the quicker time)
e.g. coach \(A\) (52.9) > (51.5) coach \(B\) so coach \(B\) likely to be quickest
NB: “coach \(B\) since \(B\) has greater range /s.d. and so more extreme values” scores B1B1
SC: “2.2” > 3.6 (A0 in (b) and answer > 3.6)
B1: equivalent reasons as above but choose \(A\)
dB1: same rules as above but applied to \(A\)