The lengths of the leaves of a particular type of tree are modelled by a normal distribution. A scientist measures the lengths of a random sample of 500 leaves from this type of tree and finds that 42 are less than 4 cm long and 100 are more than 10 cm long.

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

1. [5] B1 1.378 < z₁≤ 1.379 or –1.379 < z₁ < -1.378
2. B1 0.841 < Z2 ≤ 0.842 or -0.842 < z2 ≤ −0.841
3. M1 Use of standardisation formula once with μ, σ, a z-value and 4 or 10, allow continuity correction, not σ² or √σ
4. M1 Use either the elimination method or the substitution method to solve two equations in µ and σ.
5. A1 2.70 < σ≤2.71 7.72 <μ≤7.73
6. [4] M1 Identifying 2 and -2 as the appropriate z-values
7. B1 Calculating the appropriate area from stated phis of z-values which must be ± the same number
8. A1 Accept AWRT 0.954
9. B1 FT their 4SF (or better) probability, final answer must be positive integer