A-LevelMathematicsProbability and statistics9709/61 · Oct/Nov 2023 · Q310 Marks

A website owner finds that, on average, his website receives 0.3 hits per minute. He believes that the number of hits per minute follows a Poisson distribution.

(a)(i)Assume that the owner is correct. Find the probability that there will be at least 4 hits during a 10-minute period.[3]
(a)(ii)Use a suitable approximating distribution to find the probability that there will be fewer than 40 hits during a 3-hour period.[4]
(b)(i)Explain why this fact contradicts the owner's belief that the number of hits per minute follows a Poisson distribution.[1]
(b)(ii)Specify separate Poisson distributions that might be suitable models for the number of hits during the day-time and during the night-time.[2]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[B1] For mean = 3.
[M1] Any λ. Allow one end error.
[A1] No working scores B1.
[M1] soi
[M1] Allow with wrong or no continuity correction. For standardising with their mean and variance.
[M1] For area consistent with their working.
[A1] Special case: if no working seen, 0.0242 scores SC B3, 0.0284 scores SC B2.
[B1] 'Mean not constant' or' ‘number of hits per minute not constant' or 'not a constant rate'
[M1] May be implied by answer.
[A1] Accept Po(24) [per daytime hour], Po(12) [per night time hour]. Accept Po(288) [per day time shift], Po(144)[ per night time shift]. Note: Po(432), Po(216) scores M0A0.

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