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    HLPaper 2

    The durations taken for male participants to finish a marathon can be represented by a normal distribution with a mean of 196 minutes and a standard deviation of 24 minutes.

    It is observed that 5% of the male participants finish the marathon in less than T1{T_1}T1​ minutes.

    The durations taken for female participants to finish the marathon can be represented by a normal distribution with a mean of 210 minutes. It is observed that 58% of female participants finish the marathon between 185 and 235 minutes.

    1.

    Find the probability that a participant selected at random will finish the marathon in less than 3 hours.

    [2]
    Verified
    Solution
    • This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure. T∼N(196,242)T \sim N(196,{24^2})T∼N(196,242) P(T < 180) = 0.252 (M1)A1 [2 marks]
    2.

    Calculate T1T_1T1​.

    [2]
    Verified
    Solution

    P(T<T1)=0.05P(T < T_1) = 0.05P(T<T1​)=0.05 (M1) T1=157T_1 = 157T1​=157 A1 [2 marks]

    3.

    Find the standard deviation of the durations taken by female participants.

    [4]
    Verified
    Solution

    F∼N(210,σ2)F \sim N(210,{\sigma ^2})F∼N(210,σ2)

    P(F<235)=0.79\text{P}(F < 235) = 0.79P(F<235)=0.79 (M1)

    235−210σ=0.806421\frac{{235 - 210}}{\sigma} = 0.806421σ235−210​=0.806421 or equivalent (M1)(A1)

    σ=31.0\sigma = 31.0σ=31.0 A1

    [4 marks]

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