AS June 2025 Q3

EdexcelAS paperCurrent spec11 marksIncludes hypothesis testingPoisson Distribution

3. Raoul, Steffi and Taro are catching butterflies for research.

The number of butterflies caught by Raoul per hour may be assumed to follow a Poisson distribution with mean 4.2

Find the probability that Raoul catches

(a)

(i) exactly 6 butterflies in a randomly selected one-hour period, (1)

(ii) exactly 1 butterfly in a randomly selected 10-minute period. (2)

Following a long period without rain, Raoul believes there will now be a change to the rate at which he catches butterflies.
To test his belief, he uses the random variable \(R\) to represent the number of butterflies he catches in a 4-hour period.

A hypothesis test is to be carried out to determine whether or not there is support for Raoul’s belief. The null hypothesis of the test will be rejected if \(R \leqslant 9\) or \(R \gt 25\)

(b) Stating the hypotheses clearly, find the actual level of significance of the test. (4)

The number of butterflies caught by Steffi per hour may be assumed to follow a Poisson distribution with mean 3.2

(c) Find the probability that in exactly 2 of the next 4 hours, Steffi catches less than or equal to 3 butterflies each hour. (3)

The number of butterflies caught by Taro per hour may be assumed to follow a Poisson distribution with mean 2.7

Steffi and Taro both go to catch butterflies in a field one day.

Taro models the total number of butterflies caught per hour with a Poisson distribution with mean 3.2 + 2.7 = 5.9

(d) State a condition that would be needed for Taro’s model to be valid. (1)

Mark scheme (a)

Scheme	Marks	AO
(i) \(\mathrm{P}(X = 6) = 0.11432\ldots\) awrt 0.114	B1	1.1b
	(1)
(ii) \(Y \sim \mathrm{Po}(0.7)\)	M1	3.3
\(\mathrm{P}(Y = 1) = 0.34760\ldots\) awrt 0.348	A1	1.1b
	(2)

Notes

(i) B1: awrt 0.114

(ii) M1: Writing or using \(\mathrm{Po}(0.7)\) model

A1: awrt 0.348 correct answer scores 2 out of 2

Mark scheme (b)

Scheme	Marks	AO
\(\mathrm{H}_0: \lambda = 4.2 \qquad \mathrm{H}_1: \lambda \neq 4.2\) (allow 16.8)	B1	2.5
\(R \sim \mathrm{Po}(16.8)\)	M1	3.3
\(\mathrm{P}(R \leqslant 9) = 0.02896\ldots\) \(\mathrm{P}(R \gt 25)\ [= 1 - \mathrm{P}(R \leqslant 25) = 1 - 0.97769\ldots] = 0.02230\ldots\)	A1	3.4
Actual level of significance [\(= 0.02896\ldots + 0.02230\ldots\)] = awrt 0.0513	A1	1.1b
	(4)

Notes

B1: Both hypotheses correct in terms of \(\lambda\) or \(\mu\) Allow 4.2 or 16.8

M1: Writing or using a \(\mathrm{Po}(16.8)\) model
(may be implied by awrt 0.029 or awrt 0.022 or awrt 0.98)

A1: Either correct tail probability awrt 0.029 or awrt 0.022

A1: Correct level of significance awrt 0.0513 (allow equivalent percentage but isw once a correct answer is seen)

Mark scheme (c)

Scheme	Marks	AO
\(\mathrm{P}(S \leqslant 3) = 0.6025\ldots\)	B1	1.1b
\(J \sim \mathrm{B}(4, \text{“}0.6025\ldots\text{”})\) or \(6 \times (\text{“}0.6025\text{”})^2(1 - \text{“}0.6025\text{”})^2\)	M1	3.3
\(\mathrm{P}(J = 2) = 0.34413\ldots\) awrt 0.344	A1	1.1b
	(3)

Notes

B1: awrt 0.603 (may be implied by a correct answer awrt 0.344)

M1: Writing or using \(\mathrm{B}(4, \text{“}0.6025\ldots\text{”})\) where “their 0.6025…” must be a probability

A1: awrt 0.344

Mark scheme (d)

Scheme	Marks	AO
Only valid if they are catching butterflies independently of each other	B1	3.5b
	(1)
	(11 marks)

Notes

B1: A correct comment in context on the validity of the model which must include underlined words or equivalent
Ignore extraneous non-contradictory comments.