AS June 2019 Q1

EdexcelAS paperCurrent spec10 marksIncludes hypothesis testingCorrelation

1. Bara is investigating whether or not the two judges of a skating competition are in agreement. The two judges gave a score to each of the 8 skaters in the competition as shown in the table below.

	Skater
	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Judge 1	71	70	72	62	63	61	57	53
Judge 2	73	71	67	64	62	56	52	53

Bara decided to calculate Spearman’s rank correlation coefficient for these data.

(a) Calculate Spearman’s rank correlation coefficient between the ranks of the two judges. (4)

(b) Test, at the 1% level of significance, whether or not the two judges are in agreement. (4)

Judge 1 accidentally swapped the scores for skaters \(D\) and \(E\). The score for skater \(D\) should be 63 and the score for skater \(E\) should be 62

(c) Without carrying out any further calculations, explain how Spearman’s rank correlation coefficient will change. Give a reason for your answer. (2)

Mark scheme (a)

Scheme

Marks

Skater Judge	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
1	2	3	1	5	4	6	7	8
2	1	2	3	4	5	6	8	7
\(d^2\)	1	1	4	1	1	0	1	1

M1
dM1

1.1b
1.1b

\(\sum d^2 = 10\)

\(r_s = 1 - \dfrac{6 \times \text{“}10\text{”}}{8(64 - 1)}\)

1.1b

\(r_s = 0.8809\ldots\) awrt 0.881

1.1b

(4)

Notes

M1: For an attempt to rank at least one row ( at least 4 correct)

dM1: dep on previous M mark being awarded. For an attempt at \(d\) or \(d^2\) row for their ranks.

M1: for use of \(1 - \dfrac{6 \times \text{“their} \sum d^2\text{”}}{8(64 - 1)}\) Allow if not ranked \(\sum d^2 = 85\)

A1: awrt 0.881

Mark scheme (b)

Scheme	Marks	AO
\(\mathrm{H}_0 : \rho = 0 \quad \mathrm{H}_1 : \rho \gt 0\)	B1	2.5
Critical Value \(\rho = 0.8333\)	B1	1.1b
\(r_s = 0.8809\) lies in the critical region/reject \(\mathrm{H}_0\)/significant	M1	2.1
There is evidence that the two judges are in agreement.	A1cso	2.2b
	(4)

Notes

B1: Both hypotheses stated in terms of \(\rho\)

B1: for correct critical value. Allow even if 2 tail test (sign must match their \(r_s\))

M1: for comparing their 0.881 with “their 0.8333”

A1cso: All previous marks awarded. For a correct contextual conclusion with no contradictions seen

Mark scheme (c)

Scheme	Marks	AO
The \(\sum d^2\) will decrease since the new rankings given by Judge 1 are now the same as the rankings given by Judge 2 for Skater D and E whereas previously they were different	M1	2.4
therefore Spearman’s rank correlation coefficient will increase	A1	2.2a
	(2)
	(10 marks)

Notes

M1: For a correct explanation to support their answer given.
\(\sum d^2\) decreases or \(d/d^2\) decreases for \(D\) and \(E\)
and idea of same rankings eg \(d^2\) will reduce by 2 . Do not allow \(d^2\) will reduce by 1

A1: for a correct deduction from the information. Allow closer to 1.