AS June 2018 Q2

EdexcelAS paperCurrent spec11 marksIncludes hypothesis testingPoisson Distribution

2. The number of heaters, \(H\), bought during one day from Warmup supermarket can be modelled by a Poisson distribution with mean 0.7

(a) Calculate \(\mathrm{P}(H \geqslant 2)\) (1)

The number of heaters, \(G\), bought during one day from Pumraw supermarket can be modelled by a Poisson distribution with mean 3, where \(G\) and \(H\) are independent.

(b) Show that the probability that a total of fewer than 4 heaters are bought from these two supermarkets in a day is 0.494 to 3 decimal places. (2)

(c) Calculate the probability that a total of fewer than 4 heaters are bought from these two supermarkets on at least 5 out of 6 randomly chosen days. (3)

December was particularly cold. Two days in December were selected at random and the total number of heaters bought from these two supermarkets was found to be 14

(d) Test whether or not the mean of the total number of heaters bought from these two supermarkets had increased. Use a 5% level of significance and state your hypotheses clearly. (5)

Scheme	Marks	AO
\(\mathrm{P}(H \geqslant 2) = 0.1558\) awrt 0.156	B1	1.1b
	(1)

Mark scheme (b)

Scheme	Marks	AO
\(H \sim \mathrm{Po}(0.7) \qquad G \sim \mathrm{Po}(3)\)
\(Y = H + G \rightarrow Y \sim \mathrm{Po}(3.7)\)	M1	3.4
\(\mathrm{P}(Y \leqslant 3) = 0.494\)*	A1cso*	1.1b
	(2)

Notes

M1: For combining distributions and use of \(\mathrm{Po}(3.7)\)

A1*cso: \(\mathrm{P}(Y \leqslant 3) = 0.494\) we need to see \(\mathrm{P}(Y \leqslant 3)\) or \(\mathrm{P}(Y \lt 4)\) allow different letters.

Mark scheme (c)

Scheme	Marks	AO
\(K \sim \mathrm{B}(6, 0.494)\)	M1	3.3
\(\mathrm{P}(K \geqslant 5) = 1 - \mathrm{P}(K \leqslant 4)\)	M1	1.1b
\(= 1 - 0.896\ldots\)
\(= 0.1039\ldots\) awrt 0.104	A1	1.1b
	(3)

Notes

M1: Setting up a new model \(\mathrm{B}(6, 0.494)\) may be implied by a correct answer or \({}^6C_n (0.494)^n (0.506)^{6-n}\)

M1: Using \(1 - \mathrm{P}(K \leqslant 4)\)

A1: awrt 0.104

Mark scheme (d)

Scheme	Marks	AO
\(\mathrm{H}_0: \lambda = \text{“}3.7\text{”} \qquad \mathrm{H}_1: \lambda \gt \text{“}3.7\text{”}\)	B1ft	2.5
\(J \sim \mathrm{Po}(7.4)\)	B1ft	1.1b
Method 1: \(\mathrm{P}(J \geqslant 14) = 1 - \mathrm{P}(J \leqslant 13) = 1 - 0.9804\ldots\) Method 2: \(\mathrm{P}(J \geqslant 12) = 0.0735\ldots\) \(\mathrm{P}(J \geqslant 13) = 0.0391\ldots\)	M1	1.1b
Method 1: \(= 0.0195\ldots\) Method 2: CR \(J \geqslant 13\)	A1	1.1b
\(0.0195 \lt 0.05\) or \(14 \geqslant 13\) or 14 is in the critical region or 14 is significant or Reject \(\mathrm{H}_0\). There is evidence at the 5% level of significance that the number of heaters brought in total from the two supermarkets has increased.	A1	2.2b
	(5)
	(11 marks)

Notes

B1: Both hypotheses correct using \(\lambda\) or \(\mu\). ft \(\text{“}3.7\text{”}\) from their 3.7 in part (b) and allow \(2 \times \text{“their 3.7”}\) Ignore any words

B1: Realising that \(\mathrm{Po}(2 \times \text{“their 3.7”})\) is to be used. This may be stated or used.

M1: writing or using \(1 - \mathrm{P}(J \leqslant 13)\) or \(1 - \mathrm{P}(J \lt 14)\)
or if finding a CR for writing \(\mathrm{P}(J \geqslant 12) = 0.0735\ldots\) and \(\mathrm{P}(J \geqslant 13) = 0.0391\ldots\)

A1: awrt 0.0195 or CR \(J \geqslant 13\) or \(J \gt 12\)

A1: A fully correct solution and drawing a correct inference in context.