A2 October 2020 Q1

EdexcelCurrent spec13 marksIncludes hypothesis testingPoisson Distribution Quality of Tests

1. The number of customers entering Jeff’s supermarket each morning follows a Poisson distribution.

Past information shows that customers enter at an average rate of 2 every 5 minutes.

Using this information,

(a)

(i) find the probability that exactly 26 customers enter Jeff’s supermarket during a randomly selected 1-hour period one morning, (2)

(ii) find the probability that at least 21 customers enter Jeff’s supermarket during a randomly selected 1-hour period one morning. (2)

A rival supermarket is opened nearby. Following its opening, the number of customers entering Jeff’s supermarket over a randomly selected 40-minute period is found to be 10

(b) Test, at the 5% significance level, whether or not there is evidence of a decrease in the rate of customers entering Jeff’s supermarket. State your hypotheses clearly. (4)

A further randomly selected 20-minute period is observed and the hypothesis test is repeated.
Given that the true rate of customers entering Jeff’s supermarket is now 1 every 5 minutes,

Mark scheme (a)

Scheme	Marks	AO
(i) \(X \sim \mathrm{Po}(24)\)	B1	3.4
\(\mathrm{P}(X = 26) = 0.071912\ldots\) awrt 0.0719	B1	1.1b
	(2)
(ii) \(\mathrm{P}(X \geqslant 21) = 1 - \mathrm{P}(X \leqslant 20)\ [= 1 - 0.24263\ldots]\)	M1	3.4
\(= 0.75736\ldots\) awrt 0.757	A1	1.1b
	(2)

Notes

(i) B1: For realising the distribution is \(\mathrm{Po}(24)\) (May be seen or implied in part (ii))

B1: awrt 0.0719

(ii) M1: Writing or using \(1 - \mathrm{P}(X \leqslant 20)\)

A1: awrt 0.757

Mark scheme (b)

Scheme	Marks	AO
\(\mathrm{H}_0: \lambda = 2 \qquad [\mu = 16]\) \(\mathrm{H}_1: \lambda \lt 2 \qquad [\mu \lt 16]\)	B1	2.5
\(\mathrm{P}(Y \leqslant 10 \mid Y \sim \mathrm{Po}(16)) = 0.077396\ldots\) awrt 0.0774	B1	1.1b
Not significant / Do not reject \(\mathrm{H}_0\) / 10 is not in the CR	M1	1.1b
There is not sufficient evidence to suggest a decrease/change in the rate of customers entering Jeff’s supermarket.	A1	2.2b
	(4)

Notes

B1: Both hypotheses correct (must use \(\mu\) or \(\lambda\))

B1: awrt 0.0774 Allow awrt 0.08 from a correct probability statement.
allow CR: \(X \leqslant 9\)

M1: Correct non-contextual conclusion (may be implied by correct contextual conclusion). Allow a f.t. comparison of ‘their \(p\)’ with 0.05
(Ignore any contradictory contextual comments for this mark)

A1: A fully correct solution drawing a correct inference in context with all previous marks in (b) scored.

Mark scheme (c)

Scheme	Marks	AO
Use of \(\mathrm{Po}(8)\) to attempt critical region	M1	2.1
Critical region is \(Y \leqslant 3\) / \(\mathrm{H}_0\) is not rejected when \(Y \geqslant 4\)	A1	1.1b
True distribution is \(W \sim \mathrm{Po}(4)\)	B1	2.1
\(\mathrm{P}(W \geqslant 4 \mid W \sim \mathrm{Po}(4)) = 1 - \mathrm{P}(W \leqslant 3)\ [= 1 - 0.43347\ldots]\)	M1	1.1b
\(= 0.56652\ldots\) awrt 0.567	A1	1.1b
	(5)
	(13 marks)

Notes

M1: Use of \(\mathrm{Po}(8)\) to attempt critical region [\(\mathrm{P}(Y \leqslant 3) = 0.0423..\ \ \mathrm{P}(Y \leqslant 4) = 0.0996..\)]

A1: Finding critical region for the test \(Y \leqslant 3\) which must come from \(\mathrm{Po}(8)\).

B1: Identifying the need to use \(\mathrm{Po}(4)\) as the true distribution.
Allow \(\mathrm{Po}(4)\) seen or used for this mark.

M1: Writing or using \(\mathrm{P}(W \geqslant \text{‘}4\text{’})\) or \(1 - \mathrm{P}(W \leqslant \text{‘}3\text{’})\) from \(\mathrm{Po}(4)\). Allow f.t. on their identified CR but must be using \(\mathrm{Po}(4)\)

A1: awrt 0.567