S4 June 2013 (R) Q3

EdexcelOld spec10 marksIncludes hypothesis testingPoisson Distribution Quality of Tests

3. The number of houses sold per week by a firm of estate agents follows a Poisson distribution with mean 2. The firm believes that the appointment of a new salesman will increase the number of houses sold. The firm tests its belief by recording the number of houses sold, \(x\), in the week following the appointment. The firm sets up the hypotheses \(\mathrm{H}_0 : \lambda = 2\) and \(\mathrm{H}_1 : \lambda \gt 2\), where \(\lambda\) is the mean number of houses sold per week, and rejects the null hypothesis if \(x \geqslant 3\)

(a) Find the size of the test. (2)

(b) Show that the power function for this test is \[1 - \frac{1}{2}\mathrm{e}^{-\lambda}(2 + 2\lambda + \lambda^2)\] (3)

The table below gives the values of the power function to 2 decimal places.

\(\lambda\)	2.5	3.0	3.5	4.0	5.0	7.0
Power	0.46	\(r\)	0.68	\(s\)	0.88	0.97

Table 1

(d) Draw a graph of the power function. (2)

(e) Find the range of values of \(\lambda\) for which the power of this test is greater than 0.6 (1)

Mark scheme (a)

Scheme	Marks
\(X \sim \mathrm{Po}(2)\)
Size \(= \mathrm{P}(X \geqslant 3 / \lambda = 2)\)
\(= 1 - 0.6767\)	M1
\(= 0.3233\) awrt 0.323	A1
	(2)

Notes

M1 for correct expression for size using Po(2)

Mark scheme (b)

Scheme	Marks
Power \(= 1 - \mathrm{P}(0) - \mathrm{P}(1) - \mathrm{P}(2)\)	M1
\(= 1 - \mathrm{e}^{-\lambda} - \lambda\mathrm{e}^{-\lambda} - \dfrac{\lambda^2\mathrm{e}^{-\lambda}}{2!}\)	A1
\(= 1 - \dfrac{1}{2}\mathrm{e}^{-\lambda}\left(2 + 2\lambda + \lambda^2\right)\)	A1 cso
	(3)

Notes

1st M1 for a correct expression in terms of probabilities. Allow \(1 - \mathrm{P}(X \leqslant 2)\) or \(1 - \mathrm{P}(X \lt 3)\)

1st A1 for correct equation in \(\lambda\)

2nd A1 cso

Scheme	Marks
\(\lambda \gt 3.1\) allow numbers in range 3.1–3.2	B1
	(1)
	(10 marks)

Scheme	Marks
\(r = 0.58 \qquad s = 0.76\)	B1, B1
	(2)

Scheme	Marks
	B1ft points B1ft curve
	(2)