A-LevelMathematicsProbability and statistics9709/63 · May/June 2018 · Q68 Marks

The diameters of apples in an orchard have a normal distribution with mean 5.7cm and standard deviation 0.8 cm. Apples with diameters between 4.1 cm and 5 cm can be used as toffee apples.

(i)Find the probability that an apple selected at random can be used as a toffee apple.[3]
(ii)250 apples are chosen at random. Use a suitable approximation to find the probability that fewer than 50 can be used as toffee apples.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.

[3] M1 At least one standardising no cc no sq rt no sq using 5.7 and 0.8 and either 4.1 or 5
[3] M1 Correct area Φ – Φ legitimately obtained – need 2 negative z-values or 2 positives - not one of each
[3] A1 Correct final answer
[5] B1ft Correct unsimplified mean and var ft their prob for (i) providing (0 <p<1) Implied by σ = √34.944 =5.911
[5] M1 ± Standardising using 50, their mean and sd; must have sq rt.
[5] M1 49.5 or 50.5 seen as a cc
[5] M1 Correct area Φ(> 0.5 for + z and < 0.5 for –z)in their final answer
[5] A1 Correct final answer

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Topic: Probability and statistics, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source: A-Level Mathematics, May/June 2018 Paper 6 Variant 3, question 6.

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