A-LevelMathematicsProbability and statistics9709/61 · May/June 2023 · Q57 Marks

Large packets of rice are packed in cartons, each containing 20 randomly chosen packets. The masses of these packets are normally distributed with mean 1010g and standard deviation 3.4 g. The masses of the cartons, when empty, are independently normally distributed with mean 50g and standard deviation 2.0 g.

(a)Find the variance of the masses of full cartons.[2]
(b)Small packets of rice are packed in boxes. The total masses of full boxes are normally distributed with mean 6730 g and standard deviation 15.0g. The masses of the boxes and cartons are distributed independently of each other. Find the probability that the mass of a randomly chosen full carton is more than three times the mass of a randomly chosen full box.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[M1] 2.0^2 + 20 × 3.4^2
[A1] = 235.2
[B1] E(C-3B) = 50 + 20×1010 -3×6730 or 60
[M1] Var(C-3B) = '235.2' + 9×15^2 or 2260.2 (FT their values from (a).)
[M1] (0-60) / √2260.2 (Standardising with their values (could be implied). [= -1.262])
[M1] 1 – Φ(‘−1.262') = Φ(‘1.262') (Probability area consistent with their values.)
[A1] = 0.897 (3 sf)

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