| Date | May 2017 | Marks available | 4 | Reference code | 17M.2.hl.TZ1.9 |
| Level | HL only | Paper | 2 | Time zone | TZ1 |
| Command term | Find | Question number | 9 | Adapted from | N/A |
Question
The times taken for male runners to complete a marathon can be modelled by a normal distribution with a mean 196 minutes and a standard deviation 24 minutes.
It is found that 5% of the male runners complete the marathon in less than \({T_1}\) minutes.
The times taken for female runners to complete the marathon can be modelled by a normal distribution with a mean 210 minutes. It is found that 58% of female runners complete the marathon between 185 and 235 minutes.
Find the probability that a runner selected at random will complete the marathon in less than 3 hours.
Calculate \({T_1}\).
Find the standard deviation of the times taken by female runners.
Markscheme
\(T \sim N(196,{\text{ }}{24^2})\)
\({\text{P}}(T < 180) = 0.252\) (M1)A1
[2 marks]
\({\text{P}}(T < {T_1}) = 0.05\) (M1)
\({T_1} = 157\) A1
[2 marks]
\(F \sim N(210,{\text{ }}{\sigma ^2})\)
\({\text{P}}(F < 235) = 0.79\) (M1)
\(\frac{{235 - 210}}{\sigma } = 0.806421\) or equivalent (M1)(A1)
\(\sigma = 31.0\) A1
[4 marks]
