AS October 2020 Q4

EdexcelAS paperCurrent spec8 marksIncludes hypothesis testingPoisson Distribution

4. During the morning, the number of cyclists passing a particular point on a cycle path in a 10-minute interval travelling eastbound can be modelled by a Poisson distribution with mean 8

The number of cyclists passing the same point in a 10-minute interval travelling westbound can be modelled by a Poisson distribution with mean 3

(a) Suggest a model for the total number of cyclists passing the point on the cycle path in a 10-minute interval, stating a necessary assumption. (2)

Given that exactly 12 cyclists pass the point in a 10-minute interval,

(b) find the probability that at least 11 are travelling eastbound. (3)

After some roadworks were completed, the total number of cyclists passing the point in a randomly selected 20-minute interval one morning is found to be 14

(c) Test, at the 5% level of significance, whether there is evidence of a decrease in the rate of cyclists passing the point.
State your hypotheses clearly. (3)

Mark scheme (a)

Scheme	Marks	AO
[\(X \sim \mathrm{Po}(8) \qquad Y \sim \mathrm{Po}(3)\)] [\(X + Y \sim\)] \(\mathrm{Po}(11)\)	B1	3.3
The number of cyclists travelling eastbound is independent of the number of cyclists travelling westbound.	B1	3.5b
	(2)

Notes

B1: Correct model

B1: Correct modelling assumption in context (must mention cyclists oe)

Mark scheme (b)

Scheme	Marks	AO
\(\dfrac{\mathrm{P}(X = 11) \times \mathrm{P}(Y = 1) + \mathrm{P}(X = 12) \times \mathrm{P}(Y = 0)}{\mathrm{P}(X + Y = 12)}\)	M1 M1	2.1 1.1b
\(= 0.1204\ldots\) awrt 0.120	A1	1.1b
	(3)

Notes

M1: Attempt at ratio expression with denominator \(\mathrm{P}(X + Y = 12)\) (may see \(0.10942\ldots\))

M1: Probability expression for numerator (may be implied by \(0.01317\ldots\))

A1: awrt 0.120 accept 0.12 with correct working seen

Alternative use of binomial:

M1: Use of \(C \sim \mathrm{B}(12, \tfrac{8}{11})\)

M1: \(\mathrm{P}(C \geqslant 11) = 1 - \mathrm{P}(C \leqslant 10)\)

A1: awrt 0.120 accept 0.12 with correct working seen

Mark scheme (c)

Scheme	Marks	AO
\(\mathrm{H}_0: \lambda = 11 \text{ or } \mu = 22\) \(\mathrm{H}_1: \lambda \lt 11 \text{ or } \mu \lt 22\)	B1	2.5
\((E + W) \sim \mathrm{Po}(22)\) \(\mathrm{P}(E + W \leqslant 14)\) [= awrt 0.048]	M1	3.3
(Reject \(\mathrm{H}_0\).) There is evidence that the rate(oe) of cyclists(oe) has decreased.	A1	2.2b
	(3)
	(8 marks)

Notes

B1: Both hypotheses with \(\lambda\) or \(\mu\)

M1: Using \(\mathrm{Po}(22)\) to calculate \(\mathrm{P}(E + W \leqslant 14)\)

A1: A fully correct conclusion with awrt 0.048 or CR: \(E + W \leqslant 14\) drawing an inference in context.