The masses, in kilograms, of large and small sacks of grain have the distributions N(53, 11) and N(14, 3) respectively.

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [B1] E(D) = 53 - (4 × 14) = -3 OE Give at early stage
2. [B1] Var(D) = 11 + 4² × 3 [= 59] or √(11+4²×3) (= 7.68 (3 s.f.)) Give at early stage
3. [M1] (0-(-3))/√59 [= 0.391] For standardising with their values (var must be from a combination attempt) Ignore continuity correction attempts
4. [M1] 1 – Φ('0.391') For area consistent with their values
5. [A1] 0.348 (3 s.f.) As final answer
6. [B1] E(T) = 12 × 53 + 25 × 14 [= 986] Give at early stage (N.B. accept E(T– 1000)=-14)
7. [B1] Var(T) = 12 × 11 + 25 × 3 [= 207] Or √(12×11+25×3) (= 14.4 (3sf)) Give at early stage
8. [M1] (1000-986)/√207 [= 0.973] For standardising with their values (var must be from a combination attempt) Ignore continuity correction attempts
9. [M1] Φ('0.973') For area consistent with their values
10. [A1] 0.835 (3 sf) As final answer