A-LevelMathematicsProbability and statistics9709/51 · May/June 2020 · Q69 Marks

The lengths of female snakes of a particular species are normally distributed with mean 54 cm and standard deviation 6.1 cm.

(a)Find the probability that a randomly chosen female snake of this species has length between 50 cm and 60 cm.[4]
(b)The lengths of male snakes of this species also have a normal distribution. A scientist measures the lengths of a random sample of 200 male snakes of this species. He finds that 32 have lengths less than 45 cm and 17 have lengths more than 56 cm. Find estimates for the mean and standard deviation of the lengths of male snakes of this species.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

[1] P( (50-54)/6.1 < Z < (60-54)/6.1 ) = P(-0.6557 < Z < 0.9836)
[1] Both values correct
[1] Φ(0.9836) – Φ(−0.6557) = Φ(0.9836) + Φ(0.6557) – 1 = 0.8375+0.7441-1 (Correct area)
[1] 0.582
[1] One appropriate standardisation equation with μ, σ, z-value (not probability) and 45 or 56.
[1] (45-μ)/σ = -0.994
[1] (56-μ)/σ = 1.372
[1] 11 = 2.366 σ (For correct algebraic elimination of μ or σ from their two simultaneous equations to form an equation in one variable)
[1] σ=4.65, μ=49.6

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