Farmer Jones grows apples. The weights, in grams, of the apples grown this year are normally distributed with mean 170 and standard deviation 25. Apples that weigh between 142 grams and 205 grams are sold to a supermarket.

Mark scheme

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

1. [M1] Use of ± standardisation formula once substituting 170, 25 and either 142 or 205 appropriately.. Condone 25² and continuity correction ±0.5.
2. [A1] Both correct. Accept unsimplified.
3. [M1] Calculating the appropriate area from stated phis of z-values.
4. [A1] AWRT, not from wrong working
5. [B1 FT] P(X > 205) = 1 – 0.9192 = 0.0808. Correct or FT from part 5(a).
6. [M1] Correct or their 0.0808 × 0.30 × k + their 0.788 × 0.24 × k, k positive integer.
7. [A1] [$]4266.24. 4265 < income < 4270, not from wrong working
8. [B1] 0.5828 < z < 0.583 or -0.583 < z < -0.5828 seen.
9. [M1] 182 and 20 substituted in ± standardisation formula, no continuity correction, not σ², √ σ, equated to a z-value.
10. [A1] 170 <= w < 170.35