A-LevelMathematicsProbability and statistics9709/63 · May/June 2025 · Q27 Marks

The lengths of pencils made at a factory are normally distributed. The standard deviation of the lengths is σcm, and the mean is supposed to be 10cm. An inspector thinks that the mean is actually greater than 10cm. He takes a random sample of 50 pencils produced at the factory and finds that the mean of these 50 lengths is 10.03 cm. He then carries out a hypothesis test.

(a)(i)Calculate the value of σ.[3]
(a)(ii)Carry out the hypothesis test at the 2.5% significance level.[3]
(b)Explain whether it was necessary to use the Central Limit Theorem in carrying out the test.[1]

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

[M1] For standardising. Must have square root 50.
[M1] Equating to 1.995 (not phi 1.995).
[A1]
[B1] Accept if seen in part (i) but not in (ii). Accept population mean but not just mean.
[M1] Valid comparison. 0.977 > 0.975 or 0.023 < 0.025. Accept 2.00 or 2.001 > 1.96.
[A1] No contradictions. In context. Not definite. Accept '. that mean length has increased' Note 2 tail test scores B0M1A0.
[B1] No, because population distribution [of lengths] is normal

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