S3 June 2016 Q3

EdexcelOld spec11 marksIncludes hypothesis testingCorrelation

(a) Describe when you would use Spearman’s rank correlation coefficient rather than the product moment correlation coefficient to measure the strength of the relationship between two variables. (1)

A shop sells sunglasses and ice cream. For one week in the summer the shopkeeper ranked the daily sales of ice cream and sunglasses. The ranks are shown in the table below.

	Sun	Mon	Tues	Weds	Thurs	Fri	Sat
Ice cream	6	4	7	5	3	2	1
Sunglasses	6	5	7	2	3	4	1

(b) Calculate Spearman’s rank correlation coefficient for these data. (3)

(c) Test, at the 5% level of significance, whether or not there is a positive correlation between sales of ice cream and sales of sunglasses. State your hypotheses clearly. (4)

The shopkeeper calculates the product moment correlation coefficient from his raw data and finds \(r = 0.65\)

(d) Using this new coefficient, test, at the 5% level of significance, whether or not there is a positive correlation between sales of ice cream and sales of sunglasses. (2)

(e) Using your answers to part (c) and part (d), comment on the nature of the relationship between sales of sunglasses and sales of ice cream. (1)

Mark scheme (a)

Scheme	Marks
Variables do not have a (joint) normal distribution Relationship is not linear The given data is ordinal Any 1	B1
	(1)

Notes

Accept ‘already ranked’ oe for ordinal
Accept one variable is not normally distributed

Mark scheme (b)

Scheme

Marks

Day	Sun	Mon	Tues	Weds	Thurs	Fri	Sat
Ice-cream	6	4	7	5	3	2	1
Sunglasses rank	6	5	7	2	3	4	1
\(d\)	0	-1	0	3	0	-2	0
\(d^2\)	0	1	0	9	0	4	0

\(\sum d^2 = 14\)

\(r_s = 1 - \dfrac{6 \times 14}{7(49 - 1)} = 0.75\)

M1A1

(3)

Notes

M1 attempt to find \(d\), \(d^2\) and sum. may be implied by sight of \(\sum d^2 = 14\)

M1 for use of the correct formula, follow through their \(\sum d^2\) if clearly stated. If answer is not correct, a correct expression is required.

A1 0.75 cao

Mark scheme (c)

Scheme	Marks
\(\mathrm{H_0}\): \(\rho = 0\), \(\mathrm{H_1}\): \(\rho \gt 0\)	B1
5% cv 0.7143	B1
Reject \(\mathrm{H_0}\)	M1
Evidence of positive correlation between sales of ice cream and sales of sunglasses.	A1cao
	(4)

Notes

1st B1 for both hypotheses in terms of \(\rho\), one tail. Allow use of \(\rho_s\).
Only award if no errors seen in hypotheses in part(c) and part(d)
Hypotheses just in words e.g. “no correlation” score B0.

B1 0.7143 cao

M1 must correctly reject / not reject the null hypothesis based on their test stat and cv oe

A1 Conclusion must mention ice cream and sunglasses

Mark scheme (d)

Scheme	Marks
(5% cv 0.6694) Accept \(\mathrm{H_0}\)	M1
Insufficient evidence of positive correlation between sales of ice cream and sales of sunglasses.	A1cao
	(2)

Notes

M1 for not rejecting / accepting null hyp

A1 must mention ice cream and sunglasses

Scheme	Marks
Suggests relationship might be non-linear.	B1
	(1)
	(11 marks)