A-LevelMathematicsProbability and statistics9709/51 · Oct/Nov 2020 · Q59 Marks

The time in hours that Davin plays on his games machine each day is normally distributed with mean 3.5 and standard deviation 0.9.

(a)Find the probability that on a randomly chosen day Davin plays on his games machine for more than 4.2 hours.[3]
(b)On 90% of days Davin plays on his games machine for more than t hours. Find the value of t.[3]
(c)Calculate an estimate for the number of days in a year (365 days) on which Davin plays on his games machine for between 2.8 and 4.2 hours.[3]

Mark scheme

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

[M1] Using ± standardisation formula, no √σ or σ², continuity correction
[M1] Appropriate area Φ, from standardisation formula P(z>...) in final solution
[A1] 0.218
[B1] ±1.282 seen (critical value)
[M1] An equation using standardisation formula with a z-value, condone √σ, σ² and continuity correction
[A1] t = 2.35 (AWRT, only dependent on M mark)
[B1FT] P(2.8 < X < 4.2) = 1 – 2 × their 5(a) OR P(2.8 < X < 4.2) = 2(Φ(0.7778) - 0.5) OR P(2.8 < X < 4.2) = Φ(0.7778) - (1 - Φ(0.7778)) = 0.5636. Accept unevaluated probability. OE. Accept 0.564.
[M1] 365 × their p
[A1FT] Accept 205 or 206, not 205.0 or 206.0, no approximation/rounding stated. FT must be an integer value.

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