A-LevelMathematicsProbability and statistics9709/63 · Oct/Nov 2018 · Q68 Marks

The lifetimes, in hours, of a particular type of light bulb are normally distributed with mean 2000 hours and standard deviation σ hours. The probability that a randomly chosen light bulb of this type has a lifetime of more than 1800 hours is 0.96. New technology has resulted in a new type of light bulb. It is found that on average one in five of these new light bulbs has a lifetime of more than 2500 hours.

(i)Find the value of σ.[3]
(ii)For a random selection of 300 of these new light bulbs, use a suitable approximate distribution to find the probability that fewer than 70 have a lifetime of more than 2500 hours.[4]
(iii)Justify the use of your approximate distribution in part (ii).[1]

Mark scheme

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

[B1] ± 1.75 seen
[M1] z = ± (1800-2000)/σ, allow cc, allow sq rt, allow sq equated to a z-value
[A1] Correct final answer www (σ=114)
[B1] Correct unsimplified mean and variance (Mean = 300 × 0.2 = 60 and variance = 300 × 0.2 × 0.8 = 48)
[M1] z = ± (x-their 60)/√their 48
[M1] 69.5 or 70.5 seen in an attempted standardisation expression as cc
[A1] Correct final answer (0.915)
[B1] Both parts evaluated are required (np = 60, nq = 240: both > 5, (so normal approximation holds))

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