S3 June 2017 Q3

EdexcelOld spec10 marksIncludes hypothesis testingCorrelation

3. A junior judge is being trained by a senior judge to learn how to assess ice skaters. After the training, the judges each assess 6 ice skaters \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\). They each list them in order of preference with the best ice skater first. The results are shown in the table below.

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

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

(b) Test, at the 5% level of significance, whether or not there is evidence of a positive correlation between the rankings of the junior judge and the senior judge. State your hypotheses clearly. (4)

Mark scheme (a)

Scheme

Marks

	Senior Judge	Junior Judge	\(d\)	\(d^2\)
\(A\)	1	3	-2	4
\(B\)	2	1	1	1
\(C\)	4	5	-1	1
\(D\)	3	2	1	1
\(E\)	6	6	0	0
\(F\)	5	4	1	1
				8

M1A1

\(\sum d^2 = 8\) or 62

\(r_s = 1 - \dfrac{6 \times 8}{6 \times 35} = \dfrac{27}{35} = 0.771\)

dM1A1

(5)

Notes

M1 for an attempt to rank judges lists ( at least 4 correct for each judge)

A1 for correct rankings for both (may be reversed). Can be implied by correct \(d^2\) or \(r_s\)

N.B. Table could be ordered in terms of Senior Judge.

A1 8 or 62 or correct \(d^2\) row

M1 for use of the correct formula, follow through their \(\sum d^2\) (Dependent on 1^st M1)
If answer is not correct, a correct expression is required.

A1 exact fraction or awrt (\(\pm\))0.771

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : \rho = 0\) \(\mathrm{H}_1 : \rho \gt 0\)	B1
Critical value \(\pm 0.8286\)	B1
(0.771<0.8286) so insufficient evidence to reject \(\mathrm{H}_0\)	M1
There is insufficient evidence to suggest a positive correlation between the judges.	A1 ft
	(4)

Notes

B1 Both hypotheses in terms of \(\rho\) or \(\rho_s\). Hypotheses just in words e.g. “no correlation” score B0

B1 Accept \(\pm 0.8857\) if 2-tailed \(\mathrm{H}_1\).

M1 Follow through their \(r_s\) and their c.v. if |cv| <1 and \(|r_s| \lt 1\)

A1 ft A correct contextualised comment that includes “judges”.

Scheme	Marks
(For positive correlation c.v.is 0.8286>0.771)
Training of junior judge was ineffective.	B1 ft
	(1)
	(10 marks)