S1 June 2016 Q6
6. The time, in minutes, taken by men to run a marathon is modelled by a normal distribution with mean 240 minutes and standard deviation 40 minutes.
Nathaniel is preparing to run a marathon. He aims to finish in the first 20% of male runners.
The time, \(W\) minutes, taken by women to run a marathon is modelled by a normal distribution with mean \(\mu\) minutes.
Given that \(\mathrm{P}(W \lt \mu + 30) = 0.82\)
| Scheme | Marks |
|---|---|
| [ \(T \sim \mathrm{N}(240, 40^2)\)…require \(\mathrm{P}(T \gt 300)\)] \(\mathrm{P}\left(Z \gt \dfrac{300 - 240}{40}\right)\) | M1 |
| \(= 1 - \mathrm{P}(Z \lt 1.5)\) or 1 – 0.9332 | M1 |
| = awrt 0.0668 or 6.68% | A1 |
| (3) |
Notes
1st M1 for standardising with 300, 240 and 40. May be implied by use of 1.5 Allow \(\pm\)
2nd M1 for 1 – P(\(Z\) < “1.5”) i.e. a correct method for finding P(\(Z\) > “1.5”)
e.g. 1 – \(p\) where \(0.5 \lt p \lt 0.99\)
A1 for awrt 0.0668 (Answer only 3/3)
| Scheme | Marks |
|---|---|
| \([\mathrm{P}(T \lt n) = 0.20 \Rightarrow]\ \ \dfrac{n - 240}{40} = -0.8416\) | M1 B1 |
| \(n\) = awrt 206 minutes | A1 |
| (3) |
Notes
M1 for an attempt to standardise with 240, 40 and \(n\) and set = \(\pm z\) (\(0.8 \lt |z| \lt 0.9\))
B1 for \(z = \pm 0.8416\) (or better) used as a \(z\) value. Do not allow for 1 − 0.8416
Calc gives 0.8416212…[May be implied by awrt 206.34, give B1 as well as A1 if seen]
A1 for awrt 206 (can be scored for using a \(z\) value of 0.84 or even 0.85)
Must follow from correct working but a range of possible \(z\) values are OK
Ans only If answer is awrt 206 score M1B0A1 (unless of course \(z = 0.8416\) seen) but awrt 206.34 scores 3/3
| Scheme | Marks |
|---|---|
| \([\mathrm{P}(W \lt \mu - 30 \mid W \lt \mu) =]\ \ \dfrac{\mathrm{P}(W \lt \mu - 30)}{\mathrm{P}(W \lt \mu)}\) | M1 |
| \(= \dfrac{1 - 0.82}{0.50}\) | A1 |
| \(= \)0.36 | A1cao |
| (3) | |
| (9 marks) |
Notes
M1 for the correct ratio expression (Not \(\mathrm{P}([W \lt 30 - \mu] \cap [W \lt \mu])\) on numerator)
Condone use of \(Z\) instead of \(W\) only if they later get a correct numerical ratio otherwise M0
However they may write \(\mathrm{P}\left(Z \lt \dfrac{-30}{\sigma}\right)\) etc which is of course fine
1st A1 for a correct numerical ratio
Use tables May see use of \(z = 0.92\) or better (calc: 0.9153650…) or \(\sigma\) = 32.6~32.8 allow:
ALT 1st M1 for \(\dfrac{\mathrm{P}(Z \lt -0.92)}{\mathrm{P}(Z \lt 0)}\) and 1st A1 for \(\dfrac{1 - 0.8212}{0.5}\) or \(\dfrac{0.1788}{0.5}\)
2nd A1 for 0.36 or an exact equivalent e.g. \(\frac{9}{25}\) (Answer only M1A1A0)
The final answer of 0.36 must come from exact values; 0.36 rounded from 0.3576 etc is A0