A-LevelMathematicsProbability and statistics9709/62 · Oct/Nov 2019 · Q69 Marks

The heights, in metres, of fir trees in a large forest have a normal distribution with mean 40 and standard deviation 8.

(i)Find the probability that a fir tree chosen at random in this forest has a height less than 45 metres.[2]
(ii)Find the probability that a fir tree chosen at random in this forest has a height within 5 metres of the mean.[2]
(iii)In another forest, the heights of another type of fir tree are modelled by a normal distribution. A scientist measures the heights of 500 randomly chosen trees of this type. He finds that 48 trees are less than 10 m high and 76 trees are more than 24 m high. Find the mean and standard deviation of the heights of trees of this type.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

[M1] + Standardise, no continuity correction, o² or √o, formula must be seen P(X<45)=P(Z < (45-40)/8) = P(Z < 0.625)
[A1] CAO 0.734(0)
[M1] Use result of part (i) or recalculated to find area OE 1-2(1-(i)) = 2(i)-1 = 2((i)−0.5)
[Alft] 0<FT from (i) < 1 or correct. 0.468
[B1] z = -1.305 (for P(X<10)=48/500=0.096)
[B1] z = 1.028 (for P(X>24)=76/500=0.152, P(X<24)=1-0.152=0.848)
[M1] Form 1 equation using 10 or 24 with μ, σ, z-value. Allow continuity correction, not σ², √σ (10-μ)/σ = -1.305 or (24-μ)/σ = 1.028
[M1] OE Solve two equations in and u to form equation in one variable 14=2.333σ
[A1] CAO, WWW σ= 6.[00], μ = 17.8[3]

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