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.

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

${\text{P}}(T < 180) = 0.252$     (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]