S3 June 2008 Q3

EdexcelOld spec14 marksIncludes hypothesis testingCorrelation

3. The product moment correlation coefficient is denoted by \(r\) and Spearman’s rank correlation coefficient is denoted by \(r_s\).

(a) Sketch separate scatter diagrams, with five points on each diagram, to show

(i) \(r = 1\),

(ii) \(r_s = -1\) but \(r \gt -1\). (3)

Two judges rank seven collie dogs in a competition. The collie dogs are labelled \(A\) to \(G\) and the rankings are as follows

Rank	1	2	3	4	5	6	7
Judge 1	\(A\)	\(C\)	\(D\)	\(B\)	\(E\)	\(F\)	\(G\)
Judge 2	\(A\)	\(B\)	\(D\)	\(C\)	\(E\)	\(G\)	\(F\)

(b)

(i) Calculate Spearman’s rank correlation coefficient for these data. (6)

(ii) Stating your hypotheses clearly, test, at the 5% level of significance, whether or not the judges are generally in agreement. (5)

Two sketches: (i) five points on a straight line of positive gradient; (ii) five points decreasing from left to right but not on a straight line — Mark scheme (a)

Scheme	Marks
	(i) B1 (ii) B1B1
	(3)

Mark scheme (b)

Scheme

Marks

(i)

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
Rank (Judge 1)	1	4	2	3	5	6	7
Rank (Judge 2)	1	2	4	3	5	7	6
\(d^2\)	0	4	4	0	0	1	1

M1M1

\(\sum d^2 = 10\)

M1A1

\(r_s = 1 - \dfrac{6 \times 10}{7 \times (49 - 1)} = 1 - \dfrac{5}{28} = \dfrac{23}{28}\) or awrt 0.821

M1A1

(6)

(ii) \(\mathrm{H}_0 : \rho = 0 \qquad \mathrm{H}_1 : \rho \gt 0\) (Allow \(\rho_S\)) (\(\mathrm{H}_1 : \rho \neq 0\) scores B0)

B1,B1

\(r_s\) 5% one tail critical value is 0.7143

Significant result or reject null hypothesis

There is evidence of a (positive) correlation between the judges or the judges agree

A1ft

(5)

(14 marks)

Notes

(i) 1^st M1 for attempting to rank one of the judges (at least 2 correct rankings)
2^nd M1 for ranking both (may be reversed) (at least 2 correct rankings)
3^rd M1 for attempting \(d^2\).
1^st A1 for \(\sum d^2 = 10\)
4^th M1 for correct use of the \(r_s\) formula

(ii) 3^rd B1 for the correct critical value – depends upon their \(\mathrm{H}_1\): \(\rho \gt 0\) needs 0.7143, \(\rho \neq 0\), 0.7857
The \(\mathrm{H}_1\) may be in words so B0B1 is possible. If no \(\mathrm{H}_1\) award for 0.7143 only.

5^th M1 for a correct statement relating their \(r_s\) and their cv (may be implied by correct comment)
3^rd A1ft follow through their \(r_s\) and their cv. Comment in context. Must mention judges.
Don’t insist on “positive” and condone it if they are using \(\rho \neq 0\).