A2 June 2019 Q8

EdexcelCurrent spec11 marksIncludes hypothesis testingCorrelation

8. Nine athletes, \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\) and \(I\), competed in both the 100 m sprint and the long jump. After the two events the positions of each athlete were recorded and Spearman’s rank correlation coefficient was calculated and found to be 0.85

(a) Stating your hypotheses clearly, test whether or not there is evidence to suggest that the higher an athlete’s position is in the 100 m sprint, the higher their position is in the long jump. Use a 5% level of significance. (4)

The piece of paper the positions were recorded on was mislaid. Although some of the athletes agreed their positions, there was some disagreement between athletes \(B\), \(C\) and \(D\) over their long jump results.

The table shows the results that are agreed to be correct.

Athlete	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)
Position in 100 m sprint	4	6	7	9	2	8	3	1	5
Position in long jump	5				4	9	3	1	2

Given that there were no tied ranks,

(b) find the correct positions of athletes \(B\), \(C\) and \(D\) in the long jump. You must show your working clearly and give reasons for your answers. (5)

(c) Without recalculating the coefficient, explain how Spearman’s rank correlation coefficient would change if athlete \(H\) was disqualified from both the 100 m sprint and the long jump. (2)

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{H}_0 : \rho_s = 0 \qquad \mathrm{H}_1 : \rho_s \gt 0\)	B1	2.5
CV = 0.6	B1	1.1b
\(r_s = 0.85\) does lie in the critical region	M1	2.1
There is evidence to suggest that there is a relationship between the position in the 100m sprint and the position in the long jump.	A1	2.2b
	(4)

Notes

B1: Both hypotheses correct written using the notation \(\rho\)

B1: awrt 0.6

M1: Drawing a correct inference using their CV and the value of \(r_s\)

A1: Drawing a correct inference in context using their CV and the value of \(r_s\)

Mark scheme (b)

Scheme	Marks	AO
\(1 - \dfrac{6\sum d^2}{9(80)} = 0.85\)	M1	3.1b
\(\sum d^2 = 18\)	A1	1.1b
\(\sum d^2\) needed is \(\text{‘}18\text{’} - 15 = 3\)	M1	1.1b
Since \(\sum d^2 = 3\) for the 3 missing places each place must contribute 1, therefore \(B\) must be in position 5 or 7. However, 5 has already been used so they must be position 7	A1	2.2a
\(C\) is 6^th and \(D\) is 8^th	A1	2.2a
SC B7, C6, D8 with no reasons B1 marks as final A1 on epen
	(5)

Notes

M1: For realising they need to equate \(1 - \dfrac{6\sum d^2}{9(80)}\) to 0.85 to enable them to find the \(\sum d^2\)

A1: 18

M1: for \(\sum d^2 = 3\)

A1: For using the information in the question with the value for \(\sum d^2\) to deduce that each must contribute 1 to the \(\sum d^2\) and explain why \(B\) must be in position 7

A1: \(C\) 6^th \(D\) 8^th

Mark scheme (c)

Scheme	Marks	AO
The \(\sum d^2\) will not change but the value of \(n\) will decrease therefore	M1	2.4
Spearman’s rank correlation will decrease	A1	2.2a
	(2)
	(11 marks)

Notes

M1: Complete explanation why it decreases

A1: using the information given to deduce that it decreases