Each year a transport firm uses X litres of gasoline and Y litres of diesel fuel, where X and Y have the independent distributions X ~ N(10700, 950²) and Y ~ N(13400, 1210²).

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.

1. [B1] E(X - Y) = 10700 – 13400 [=-2700] Oe, e.g. (Y – X).
2. [M1] Var(X - Y) = 950² + 1210² [= 2366600]
3. [M1] (0 - (their '-2700')) / sqrt(their '2366600') [= 1.755] For standardising with their E and Var.
4. [M1] 1 – Φ(their '1.755') For area consistent with their values.
5. [A1] = 0.0396 or 0.0397 (3 sf)
6. [B1] E(Total) = 10700 × 0.8 + 13400 × 0.85 [=19950]
7. [M1] Var(Total) = 950² × 0.8² + 1210² × 0.85² [= 1635412.25]
8. [M1] (22000-their '19950') / sqrt(their '1635412.25') [= 1.603] or (20000-their '19950') / sqrt(their '1635412.25') [= 0.0391] For one standardisation with their E and Var.
9. [M1] Φ(their '1.603') – Φ(their '0.0391') = 0.9455 – 0.5156 For area consistent with their values.
10. [A1] = 0.43[0] (3 sf)