S4 June 2017 Q2

EdexcelOld spec8 marksIncludes hypothesis testingPoisson Distribution Quality of Tests

2. The number of accidents per year in Daftstown follows a Poisson distribution with mean \(\lambda\). The value of \(\lambda\) has previously been 6 but Jonty claims that since the Council increased the speed limit, the value of \(\lambda\) has increased.

Jonty records the number of accidents in Daftstown in the first year after the speed limit was increased. He plans to test, at the 5% significance level, whether or not there is evidence of an increase in the mean number of accidents in Daftstown per year.

(a) Stating your hypotheses clearly, calculate the probability of a Type I error for this test. (4)

Given that there were 9 accidents in the first year after the speed limit was increased,

(b) state, giving a reason, whether or not there is evidence to support Jonty’s claim. (2)

(c) Given that the value of \(\lambda\) has actually increased to 8, calculate the probability of drawing the conclusion, using this test, that the number of accidents per year in Daftstown has not increased. (2)

Mark scheme (a)

Scheme	Marks
\(\mathrm{H}_0 : \lambda = 6,\ \mathrm{H}_1 : \lambda \gt 6\) both	B1
\(\mathrm{P}(X \geqslant 10) = 0.0839\) \(\mathrm{P}(X \geqslant 11) = 0.0426\)	M1
CR \(X \geqslant 11\)	A1
P (Type I Error) \(= 0.0426\)	A1
	(4)

Notes

B1 both hypotheses, allow use of \(\mu\)

M1 for seeing \(\left[\mathrm{P}(X \geqslant 10) =\right] 0.0839\) or \(\left[\mathrm{P}(X \geqslant 11) =\right] 0.0426\) or \(\left[\mathrm{P}(X \leqslant 9) =\right] 0.9161\) or \(\left[\mathrm{P}(X \leqslant 10) =\right] 0.9574\) oe allow a sideways slip of 1. ie 6.5/5.5

A1 for seeing \(\mathrm{P}(X \leqslant 10) = 0.9574\) or \(\mathrm{P}(X \geqslant 11) = 0.0426\) or CR \(X \geqslant 11\)

A1 0.0426

NB An answer of 0.0426 implies will get M1A1A1

Mark scheme (b)

Scheme	Marks
9 is not in the critical region therefore	M1
there is no evidence of an increase in the number of accidents per year or there is no evidence to support Jonty’s claim	A1ft
	(2)

Notes

M1 must have 9/ value oe is not in CR allow \(0.153 \gt 0.05\)

A1ft correct statement in context – need accidents or Jonty

Mark scheme (c)

Scheme	Marks
\(\lambda = 8\) \(\mathrm{P}(X \leqslant 10 \mid \lambda = 8) = 0.8159\)	M1A1
	(2)
	(8 marks)

Notes

M1 \(\mathrm{P}(X \leqslant c - 1 \mid \lambda = 8)\) with \(c - 1\) being correct or using their \(c\). Allow if a CR is stated in the form \(X \leqslant c\) for \(1 - \mathrm{P}(X \leqslant c \mid \lambda = 8)\)

A1 awrt 0.816