S3 June 2012 Q1

EdexcelOld spec12 marksIncludes hypothesis testingCorrelation

1. Interviews for a job are carried out by two managers. Candidates are given a score by each manager and the results for a random sample of 8 candidates are shown in the table below.

Candidate	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Manager \(X\)	62	56	87	54	65	15	12	10
Manager \(Y\)	54	47	71	50	49	25	30	44

(a) Calculate Spearman’s rank correlation coefficient for these data. (5)

(b) Test, at the 5% level of significance, whether there is agreement between the rankings awarded by each manager. State your hypotheses clearly. (5)

Manager \(Y\) later discovered he had miscopied his score for candidate \(D\) and it should be 54.

(c) Without carrying out any further calculations, explain how you would calculate Spearman’s rank correlation in this case. (2)

Mark scheme (a)

Scheme

Marks

\(X\)	\(Y\)	Rank \(X\)	Rank \(Y\)	\(d\)	\(d^2\)
62	54	3	2	1	1
56	47	4	5	-1	1
87	71	1	1	0	0
54	50	5	3	2	4
65	49	2	4	-2	4
15	25	6	8	-2	4
12	30	7	7	0	0
10	44	8	6	2	4

M1
M1

\(\sum d^2 = 18\)

\(r_s = 1 - \dfrac{6 \times 18}{8(64 - 1)} = 0.7857\ldots\) awrt 0.786

M1A1

(5)

Notes

1st M1 for an attempt to rank score \(X\) and score \(Y\)

2nd M1 for attempting d² for their ranks. Must be using ranks.

1st A1 for sum of 18

3rd 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 0.786

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : \rho = 0\)	B1
\(\mathrm{H}_1 : \rho \gt 0\)	B1
Critical region \(r_s \gt 0.6429\)	B1
(\(0.7857 \gt 0.6429\) sufficient evidence to) reject \(\mathrm{H}_0\)	M1
There is evidence of agreement between the scores awarded by each manager	A1ft
	(5)

Notes

1st B1 for null hypotheses in terms of \(\rho\) or \(\rho_s\)

2nd B1 for alt hyp as given

3rd B1 for cv of +0.6429 (or 0.7381 if two tailed from hyp)

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 “scores / rankings” and “manager”
Follow through their \(r_s\) and their cv (provided it is |cv| <1
Use of “association” is A0

(corrected from the printed mark scheme: the second hypothesis is printed as \(\mathrm{H}_0 : \rho \gt 0\); it is the alternative hypothesis \(\mathrm{H}_1 : \rho \gt 0\))

Mark scheme (c)

Scheme	Marks
(\(A\) and \(D\) are now) tied ranks (for Manager \(Y\))	B1
Average rank (awarded to \(A\) and \(D\)) and use \(r_s = \dfrac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}\)	B1
	(2)
	(12 marks)

Notes

1st B1 Tied ranks can be implied by 2.5, 6.5 or both 2 or 6 or description.

2nd B1 Average rank implied by 2.5 or 6.5 or description and ‘use of pmcc’.