A2 June 2019 Q5

EdexcelCurrent spec12 marksIncludes hypothesis testingPoisson Distribution Quality of Tests

5. Information was collected about accidents on the Seapron bypass. It was found that the number of accidents per month could be modelled by a Poisson distribution with mean 2.5

Following some work on the bypass, the numbers of accidents during a series of 3-month periods were recorded. The data were used to test whether or not there was a change in the mean number of accidents per month.

(a) Stating your hypotheses clearly and using a 5% level of significance, find the critical region for this test. You should state the probability in each tail. (5)

(b) State \(\mathrm{P}(\text{Type I error})\) using this test. (1)

Data from the series of 3-month periods are recorded for 2 years.

Given that the number of accidents per month on the bypass, after the work is completed, is actually 2.1 per month,

(d) find \(\mathrm{P}(\text{Type II error})\) for the test in part (a) (3)

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{H}_0: \lambda = 2.5\) (or \(\mu = 7.5\)) \(\mathrm{H}_1: \lambda \ne 2.5\) (or \(\mu \ne 7.5\))	B1	2.5
[\(X\) = no. of accidents in a 3-month period] \(X \sim \mathrm{Po}(7.5)\)	M1	3.3
\(\mathrm{P}(X \leqslant 2) = 0.0203\) (calc: \(0.020256\ldots\)) {or \(\mathrm{P}(X \leqslant 3) = 0.0591\)} \(\mathrm{P}(X \leqslant 13) = 0.9784\) so \(\mathrm{P}(X \geqslant 14) = 0.0216\) (calc: \(0.0215646\ldots\)) {or \(\mathrm{P}(X \geqslant 15) = 0.0103\)}	M1	3.4
Giving Critical region of: \(\boldsymbol{X \leqslant 2}\)	A1	1.1b
\(\boldsymbol{X \geqslant 14}\)	A1	1.1b
	(5)

Notes

B1 for both hypotheses in terms of \(\lambda\) or \(\mu\) (either way around)

1^st M1 for selecting the correct Po model. Sight or use of \(\mathrm{Po}(7.5)\) may be implied by 2^nd M1

2^nd M1 for using the correct model to find one of these probs with correct label (2sf or better)

1^st A1 for one end correct

2^nd A1 for a fully correct CR
Allow any letter, even CR \(\leqslant 2\) or set notation but not \(\mathrm{P}(X \leqslant 2)\)
Can have \(X \lt 3\) and \(X \gt 13\) etc

Mark scheme (b)

Scheme	Marks	AO
\([0.0203 + 0.0216] =\) awrt 0.0419 or (calc: \(0.041821366\ldots\) awrt 0.0418)	B1ft	1.2
	(1)

Notes

B1ft for awrt 0.0419 or awrt 0.0418
or ft addition of their two probs provided both are \(0 \lt \text{prob} \lt 0.025\) (awrt 3sf)

Mark scheme (c)

Scheme	Marks	AO
[Let \(M\) = no of 3-month periods with a significant result] \(M \sim \mathrm{B}(8, \text{“}0.0419\text{”})\)	M1	3.3
\([\mathrm{P}(M \geqslant 2)] = 1 - \mathrm{P}(M \leqslant 1)\)	M1	1.1b
\([= 1 - 0.9584\ldots]\) \(= 0.04153\ldots\) (calc: \(0.041394\ldots\)) [0.04139 ~ 0.04154]	A1cso	1.1b
	(3)

Notes

1^st M1 for selecting a correct binomial model, ft their answer to part (b)

2^nd M1 for a correct probability statement of \(1 - \mathrm{P}(M \leqslant 1)\) dep on a binomial selected

A1cso for answer in range [0.04139, 0.04154] dep on use of \(\mathrm{B}(8, \text{“}0.0419\text{”})\) or better

Mark scheme (d)

Scheme	Marks	AO
\(Y \sim \mathrm{Po}(6.3)\)	M1	3.3
\(\mathrm{P}(\text{Type II error}) = \mathrm{P}(3 \leqslant Y \leqslant 13)\) or \(\mathrm{P}(Y \leqslant 13) - \mathrm{P}(Y \leqslant 2)\)	M1	3.4
\([= 0.9945147\ldots - 0.049846\ldots]\) \(= 0.9446\ldots\) awrt 0.945	A1	1.1b
	(3)
	(12 marks)

Notes

1^st M1 for selecting a \(\mathrm{Po}(6.3)\) model

2^nd M1 for a correct probability statement using their Poisson model and their CR in (a) which may have just one tail.

A1 for awrt 0.945