A2 June 2022 Q3

EdexcelCurrent spec14 marksIncludes hypothesis testingPoisson Distribution

3. During the summer, mountain rescue team \(A\) receives calls for help randomly with a rate of 0.4 per day.

(a) Find the probability that during the summer, mountain rescue team \(A\) receives at least 19 calls for help in 28 randomly selected days. (2)

The leader of mountain rescue team \(A\) randomly selects 250 summer days from the last few years.
She records the number of calls for help received on each of these days.

(b) Using a Poisson approximation, estimate the probability of the leader finding at least 20 of these days when more than 1 call for help was received by mountain rescue team \(A\). (4)

Mountain rescue team \(A\) believes that the number of calls for help per day is lower in the winter than in the summer. The number of calls for help received in 42 randomly selected winter days is 8

(c) Use a suitable test, at the 5% level of significance, to assess whether or not there is evidence that the number of calls for help per day is lower in the winter than in the summer. State your hypotheses clearly. (4)

During the summer, mountain rescue team \(B\) receives calls for help randomly with a rate of 0.2 per day, independently of calls to mountain rescue team \(A\).

The random variable \(C\) is the total number of calls for help received by mountain rescue teams \(A\) and \(B\) during a period of \(n\) days in the summer.
On a Monday in the summer, mountain rescue teams \(A\) and \(B\) each receive a call for help.

Given that over the next \(n\) days \(\mathrm{P}(C = 0) \lt 0.001\)

(d) calculate the minimum value of \(n\) (3)

(e) Write down an assumption that needs to be made for the model to be appropriate. (1)

Mark scheme (a)

Scheme	Marks	AO
\(W \sim \mathrm{Po}(11.2)\) and \(\mathrm{P}(W \geqslant 19) = 1 - \mathrm{P}(W \leqslant 18)\) or suitable 3sf probs	M1	3.4
\(\mathrm{P}(W \geqslant 19) = 0.020776\ldots\) awrt 0.021	A1	1.1b
	(2)

Notes

M1 For using the model \(\mathrm{Po}(11.2)\) implied by sight of: 0.02077… or 0.9889.. or 0.9792..

A1 awrt 0.021

Mark scheme (b)

Scheme	Marks	AO
[\(S\) = # calls per day, \(S \sim \mathrm{Po}(0.4)\)] \(\mathrm{P}(S \gt 1) = 0.061551\ldots\) awrt 0.0616	B1	1.1b
\(X \sim \mathrm{B}(250, \text{“}0.061551..\text{”})\)	M1	3.3
\(Y \sim \mathrm{Po}(\text{“}15.3879\ldots\text{”})\) [Accept \(\mathrm{Po}(15.4)\) or better] or suitable 3sf probs	M1	3.4
\(= 0.14751\ldots\) awrt 0.148	A1	1.1b
	(4)

Notes

B1 awrt 0.0616

1^st M1 Setting up a new model \(\mathrm{B}(250, \text{“}0.0616\text{”})\) [condone \(\mathrm{B}(\text{“}0.0616\text{”}, 250)\)]

2^nd M1 Seeing the model \(\mathrm{Po}(\text{their } np)\) implied by sight of: 0.1475.. or 0.89975 or 0.8524…

A1 awrt 0.148

SC if no approximation used (and 1^st M1 not seen) an answer of awrt 0.140 could get B1M1M0A0

Mark scheme (c)

Scheme	Marks	AO
\(\mathrm{H}_0: \lambda = 16.8 \qquad \mathrm{H}_1: \lambda \lt 16.8\)	B1	2.5
\(U \sim \mathrm{Po}(16.8)\)	B1	3.3
\(\mathrm{P}(U \leqslant 8) = 0.014\)	M1	1.1b
[\(0.014 \lt 0.05\) or there is sufficient evidence to reject \(\mathrm{H}_0\)] There is sufficient evidence at the 5% level of significance that the number of calls received per day is lower in winter or rate of calls is lower in winter or less calls per day in winter (o.e.)	A1	2.2b
	(4)

Notes

1^st B1 Both hypotheses correct using \(\lambda\) or \(\mu\) and 16.8 or 0.4 [Accept their ans to \(0.4 \times 42\)]

2^nd B1 Realising \(\mathrm{Po}(16.8)\) needs to be used. Sight or use of, implied by correct prob or CR

M1 For 0.014 or better (0.0141..) or CR \(X \leqslant 9\) oe must be CR and not probability.
[Allow CR \(X \leqslant 10\) with probability \(\mathrm{P}(X \leqslant 10) = 0.054\) or better]

A1 Indep of 1^st B1 (must see 2^nd B1 and M1 scored) for a correct inference in context

Mark scheme (d)

Scheme	Marks	AO
\(C \sim \mathrm{Po}(0.4 \times n + 0.2 \times n)\ [= \mathrm{Po}(0.6n)]\) or \(D \sim \mathrm{B}(n, \mathrm{e}^{-0.6}\) or awrt 0.549)	M1	3.1b
\(\mathrm{e}^{-0.6n} \lt 0.001\) or \(-0.6n \lt \ln(0.001)\) or \(n \gt 11.5\ldots\)	M1	1.1b
\(n = \underline{12}\)	A1	1.1b
	(3)

Notes

1^st M1 Selecting a suitable model. Sight of \(\mathrm{Po}(0.6n)\) or \(\mathrm{B}(n, \mathrm{e}^{-0.6})\) or implied by 2^nd M1

2^nd M1 For a correct inequality or equality involving \(n\) [Condone slips in solving]
Allow MR i.e. misread of 0.01 for 0.001 (or similar) to score M1M1A0

A1 \(n = 12\) cao [Correct answer with no incorrect working seen scores 3/3]

Mark scheme (e)

Scheme	Marks	AO
The rate of calls per day is constant or the number of calls occurring in non-overlapping time intervals is independent. or number of calls per day is independent (o.e.)	B1	2.4
	(1)
	Total 14

Notes

B1 Allow equivalent statements. Underlined words required.