The mass, in tonnes, of steel produced per day at a factory is normally distributed with mean 65.2 and standard deviation 3.6. It can be assumed that the mass of steel produced each day is independent of other days. The factory makes $50 profit on each tonne of steel produced. Find the probability that the total profit made in a randomly chosen 7-day week is less than $22000.

Mark scheme

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

1. [1] Mean of total mass (M) = 7 × 65.2 = 456.4 tonnes. OR Mean of total profit (P) = 7 × 65.2 × 50 = 22820.
2. [1] Variance of total mass (M) = 7 × 3.6² = 90.72. OR Variance of total profit (P) = 7 × 3.6² × 50² = 226800.
3. [1] Corresponding threshold for total mass: 22000/50 = 440 tonnes (if using mass method).
4. [1] Standardise the value using their mean and variance/standard deviation, e.g., z = (440 - 456.4) / sqrt(90.72) OR z = (22000 - 22820) / sqrt(226800), leading to z ≈ -1.722.
5. [1] Calculate the probability area P(Z < z_value) = 1 - Φ(|z_value|), consistent with their z-value.
6. [1] Final probability = 0.0425 or 0.0426 (accept AWRT 0.0425 or 0.0426).