S3 June 2007 Q1

EdexcelOld spec10 marksIncludes hypothesis testingCorrelation

1. During a village show, two judges, \(P\) and \(Q\), had to award a mark out of 30 to some flower displays. The marks they awarded to a random sample of 8 displays were as follows:

Display	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Judge \(P\)	25	19	21	23	28	17	16	20
Judge \(Q\)	20	9	21	13	17	14	11	15

(a) Calculate Spearman’s rank correlation coefficient for the marks awarded by the two judges. (6)

After the show, one competitor complained about the judges. She claimed that there was no positive correlation between their marks.

(b) Stating your hypotheses clearly, test whether or not this sample provides support for the competitor’s claim. Use a 5% level of significance. (4)

Mark scheme (a)

Scheme

Marks

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
\(P\) Rank	2	6	4	3	1	7	8	5
\(Q\) Rank	2	8	1	6	3	5	7	4
\(d^2\)	0	4	9	9	4	4	1	1

M1A1

\(\sum d^2 = 32\)

M1A1

\(r_S = 1 - \dfrac{6 \times 32}{8 \times (8^2 - 1)}\)

\(= \dfrac{13}{21}\) or AWRT 0.619

(6)

Notes

1^st M1 for attempting to rank both \(P\) and \(Q\).
1^st A1 for both correct (could be reversed)
2^nd M1 for attempting \(d^2\)
2^nd A1 for \(\sum d^2 = 32\).
3^rd M1 for correct use of formula for \(r_S\)

S.C. No ranking Typical answer (−3.82) can get mark for use of \(r_s\) formula and hypotheses in (b) only
(a) M0A0M0A0M1A0 (b) B1B1M0A0

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : \rho = 0 \qquad \mathrm{H}_1 : \rho \gt 0\) (\(\rho_s\) is OK) both	B1
\(r_S\) 1 tail 5% critical value is 0.6429 (Independent of their \(\mathrm{H}_1\))	B1 (\(\pm\) is OK)
\(0.619 \lt 0.6429\) or not significant	M1
So insufficient evidence of a positive correlation between judges Or competitor is justified	A1f.t.
	(4)
	(10 marks)

Notes

M1 for a correct comparison or statement about significance (o.e.)
Follow through their \(r_s\) provided \(0 \lt r_s \lt 1\)

A1f.t. for a conclusion in context. Must mention judges or marks or competitor.
If they use correlation they must say it is positive.
Follow through their positive \(r_s\) with their positive c.v. and ignore hypotheses.
So \(r_s = 0.667\) they could say competitor’s claim is not justified etc.