A-LevelMathematicsProbability and statistics9709/61 · Oct/Nov 2022 · Q48 Marks
Each month a company sells X kg of brown sugar and Y kg of white sugar, where X and Y have the independent distributions N(2500, 120²) and N(3700, 130²) respectively.
- (a)Find the mean and standard deviation of the total amount of sugar that the company sells in 3 randomly chosen months.[3]
- (b)The company makes a profit of $1.50 per kilogram of brown sugar sold and makes a loss of $0.20 per kilogram of white sugar sold. Find the probability that, in a randomly chosen month, the total profit is less than $3000.[5]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.
- [B1] Mean = [3 (2500 + 3700)] = 18600 (kg)
- [M1] Var(Total profit) = 3(120^2 + 130^2) or 93900
- [A1] sd = 306 (kg) (3 sf)
- [B1] E(1.5X-0.2Y) = 1.5x2500 – 0.20x3700 = [3010]
- [B1] Var(1.5X – 0.2Y) = 1.5^2×120^2 + 0.2^2×130^2 [= 33076]
- [M1] (3000-3010) / √their '33076' [= -0.055]
- [M1] Φ(their '-0.055') = 1 – Φ(their '0.055')
- [A1] = 0.478 (3 sf)
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