S3 June 2011 Q2

EdexcelOld spec10 marksIncludes hypothesis testingCorrelation

2. A county councillor is investigating the level of hardship, \(h\), of a town and the number of calls per 100 people to the emergency services, \(c\). He collects data for 7 randomly selected towns in the county. The results are shown in the table below.

Town	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
\(h\)	14	20	16	18	37	19	24
\(c\)	52	45	43	42	61	82	55

(a) Calculate the Spearman’s rank correlation coefficient between \(h\) and \(c\). (6)

After collecting the data, the councillor thinks there is no correlation between hardship and the number of calls to the emergency services.

(b) Test, at the 5% level of significance, the councillor’s claim. State your hypotheses clearly. (4)

Mark scheme (a)

Scheme

Marks

Town	A	B	C	D	E	F	G
\(h\) rank	1	5	2	3	7	4	6
\(c\) rank	4	3	2	1	6	7	5
\(\|d\|\)	3	2	0	2	1	3	1

\(d^2\)

\(\sum d^2 = 28\)

M1A1

\(r_s = 1 - \dfrac{6 \times 28}{7 \times 48}\)

\(= 0.5\)

(6)

Notes

1st M1 for an attempt to rank the hardship against calls

2nd M1 for attempting \(d\) for their ranks. Must be using ranks.

3rd M1 for attempting \(\sum d^2\) (must be using ranks)

1st A1 for sum of 28 (or 84)

4th M1 for use of the correct formula with their \(\sum d^2\). If answer is not correct an expression is required.

2nd A1 for awrt 0.5 (or -0.5)

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : \rho = 0, \mathrm{H}_1 : \rho \ne 0\)	B1
Critical values are \(r_s = \pm 0.7857\)	B1ft
\(0.5 \lt 0.7857\) insufficient evidence to reject \(\mathrm{H}_0\)	M1
Councillor’s claim is supported.	A1ft
	(4)
	(10 marks)

Notes

1st B1 for both hypotheses in terms of \(\rho\), \(\mathrm{H}_1\) must be two tail.

2nd B1 for cv of \(\pm\)0.7857 (or 0.7143 to ft from 1-tailed \(\mathrm{H}_1\))

M1 for a correct statement relating their \(r_s\) with their cv but cv must be such that |cv|<1

A1ft for a correct contextualised comment. Must mention “Councillor” and “claim” or “hardship” and “number of calls (to the emergency services)”
Follow through their \(r_s\) and their cv (provided it is |cv| <1
Condone use of “association” in conclusion for A1
Condone ‘positive’ in conclusion.