S2 June 2014 Q3

EdexcelOld spec13 marksIncludes hypothesis testingPoisson Distribution

3. A company claims that it receives emails at a mean rate of 2 every 5 minutes.

(a) Give two reasons why a Poisson distribution could be a suitable model for the number of emails received. (2)

(b) Using a 5% level of significance, find the critical region for a two-tailed test of the hypothesis that the mean number of emails received in a 10 minute period is 4. The probability of rejection in each tail should be as close as possible to 0.025 (2)

To test this claim, the number of emails received in a random 10 minute period was recorded.

During this period 8 emails were received.

(d) Comment on the company’s claim in the light of this value. Justify your answer. (2)

During a randomly selected 15 minutes of play in the Wimbledon Men’s Tennis Tournament final, 2 emails were received by the company.

(e) Test, at the 10% level of significance, whether or not the mean rate of emails received by the company during the Wimbledon Men’s Tennis Tournament final is lower than the mean rate received at other times. State your hypotheses clearly. (5)

Mark scheme (a)

Scheme	Marks
Any two of Emails are independent/occur at random Emails occur singly Emails occur at a constant rate	B1B1d
	(2)

Notes

B1 any correct statement with context of emails in

B1d Dependent on previous B1. Any correct statement, need not have context

SC for 2 correct statements without context B1 B0

Mark scheme (b)

Scheme	Marks
\(X \sim \mathrm{Po}(4)\)
\(\mathrm{P}(X = 0) = 0.0183\)
\(\mathrm{P}(X \geqslant 9) = 0.0214\)
CR \(X = 0; \quad X \geqslant 9\)	B1B1
	(2)

Notes

B1 \(X = 0\) or \(X \leqslant 0\) Allow any letter.

B1 \(X \geqslant 9\) or \(X \gt 8\) Allow any letter.

SC if write correct CR’s as probability statements award B1 B0

For these 2 marks ignore any union sign (\(\cup\)) or intersection sign (\(\cap\))

Mark scheme (c)

Scheme	Marks
\(0.0183 + 0.0214 = 0.0397\) or 3.97%	M1A1
	(2)

Notes

M1 adding their probabilities of ‘their’ critical regions if sum gives a probability less than 1 or award if a correct answer given

A1 awrt 0.0397

Mark scheme (d)

Scheme	Marks
8 is not in the critical region or \(\mathrm{P}(X \geqslant 8) = 0.0511\)	M1
therefore there is evidence that the company’s claim is true	A1ft
	(2)

Notes

M1 correct reason ft their CR. Do not allow non-contextual contradictions.

A1 correct conclusion for their CR. Allow conclusion in context of emails are received at a rate of 2 every 5 mins

Mark scheme (e)

Scheme	Marks
\(\mathrm{H}_0 : \lambda = 6\) (or \(\lambda = 2\)) \(\qquad \mathrm{H}_1 : \lambda \lt 6\) (or \(\lambda \lt 2\)) allow \(\lambda\) or \(\mu\)	B1
Po(6)	M1
\(\mathrm{P}(X \leqslant 2) = 0.0620\) CR \(X \leqslant 2\)	A1
\(0.0620 \lt 0.10\) Reject \(\mathrm{H}_0\) or Significant.	M1 dep.
There is evidence at the 10% level of significance that the mean rate/number/amount of emails received is lower/ has decreased/is less. Or fewer emails are received	A1 cso
	(5)
	(13 marks)

Notes

(corrected from the printed mark scheme: the alternative form of \(\mathrm{H}_1\) is printed as “(or \(\lambda = 2\))”; it should be \(\lambda \lt 2\))

B1 both hypotheses correct, must have \(\lambda\) or \(\mu\) and either 2 or 6.

M1 using Po(6) may be implied by correct answer.

A1 0.062 or \(X \leqslant 2\)

M1 dependent on previous method being awarded. Do not allow conflicting non-contextual statements. Follow through their hypotheses.