A-LevelMathematicsProbability and statistics9709/62 · Oct/Nov 2021 · Q47 Marks

A certain kind of firework is supposed to last for 30 seconds, on average, after it is lit. An inspector suspects that the fireworks actually last a shorter time than this, on average. He takes a random sample of 100 fireworks of this kind. Each firework in the sample is lit and the time it lasts is noted.

(a)Give a reason why it is necessary to take a sample rather than testing all the fireworks of this kind.[1]
(b)It is given that the population standard deviation of the times that fireworks of this kind last is 5 seconds. The mean time lasted by the 100 fireworks in the sample is found to be 29 seconds. Test the inspector's suspicion at the 1% significance level.[5]
(c)State with a reason whether the Central Limit theorem was needed in the solution to part (b).[1]

Mark scheme

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

[B1] Fireworks are destroyed when tested.
[B1] Ho: Pop mean time lasted (or μ) = 30 H1: Pop mean time lasted (or μ) < 30 Not just 'mean'.
[M1] For standardising. Must have √100. Use of totals N(3000,2500) giving (2900-3000)/√2500 scores M1. No mixed methods.
[A1] ±-2
[M1] Accept -2.326 to -2.329. Valid comparison or area comparison 0.0228>0.01 or 0.9772<0.99. Accept CR method 28.837<29 or 30.163>30.
[A1 FT] In context (if used need mean or time / condone average instead of mean), not definite, e.g. not ‘mean time lasted is not less than 30 seconds'. No contradictions. Note 2 tailed test can score B0 M1 A1 M1 (comparison with 2.574–2.579) A0 (no FT).
[B1] Yes. Because population distribution is unknown [condone not Normal]. Both needed. Condone X for parent population.

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