17M.2.AHL.TZ1.H_9

pestleMathematicsAAHLPaper 217M· sl-4-9-normal-distribution-and-calculationssource ↗

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.

[2]

Calculate ${T_1}$ .

[2]

Find the standard deviation of the times taken by female runners.

[4]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$T \sim N(196,{\text{ }}{24^2})$

${\text{P}}(T < 180) = 0.252$ (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]

Examiners’ report

[N/A]