S3 June 2014 (R) Q1

EdexcelOld spec11 marksIncludes hypothesis testingCorrelation

1. A journalist is investigating factors which influence people when they buy a new car. One possible factor is fuel efficiency. The journalist randomly selects 8 car models. Each model’s annual sales and fuel efficiency, in km/litre, are shown in the table below.

Car model	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
Annual sales	1800	5400	18 100	7100	9300	4800	12 200	10 700
Fuel efficiency	5.2	18.6	14.8	13.2	18.3	11.9	16.5	17.7

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

The journalist believes that car models with higher fuel efficiency will achieve higher sales.

(b) Stating your hypotheses clearly, test whether or not the data support the journalist’s belief. Use a 5% level of significance. (4)

(d) The mean and median fuel efficiencies of the car models in the random sample are 14.5 km/litre and 15.65 km/litre respectively. Considering these statistics, as well as the distribution of the fuel efficiency data, state whether or not the data suggest that the assumption in part (c) might be true in this case. Give a reason for your answer.
(No further calculations are required.) (1)

Mark scheme (a)

Scheme

Marks

Car model	A	B	C	D	E	F	G	H
Sales rank	8	6	1	5	4	7	2	3
Fuel efficiency rank	8	1	5	6	2	7	4	3
\(d^2\)	0	25	16	1	4	0	4	0

\(\sum d^2 = 50\)

M1A1

\(r_s = 1 - \dfrac{6\sum d^2}{8(64 - 1)} = 1 - \dfrac{6 \times 50}{8 \times 63}\)

\(r_s = \dfrac{204}{504} = 0.40476\ldots\) awrt 0.405

(5)

Notes

M1 for attempting to rank at least one set of data

A1 for at least one set of data ranked correctly (NB this mark comes after 2nd M1)

M1 for attempting \(\Sigma d^2\)

M1 for correct use of formula for \(r_s\)

Mark scheme (b)

Scheme	Marks
\(\mathrm{H_0}: \rho_s = 0 \qquad \mathrm{H_1}: \rho_s \gt 0\) (accept \(\rho_s\) or \(\rho\))	B1
1 tail critical value \(\rho = 0.6429\)	B1
Test value is not in critical region so insufficient evidence to reject \(\mathrm{H_0}\) No significant evidence at 5% level to support journalist’s belief	M1A1ft
	(4)

Notes

B1 for \(\mathrm{H_0}\) and \(\mathrm{H_1}\) correct (condone \(\leqslant\) for \(\mathrm{H_0}\))

2nd B1 allow 0.7381 if their \(\mathrm{H_1}: \rho_s \neq 0\)

M1 for correct statement relating their test statistic and critical value

A1ft their test statistic, \(\mathrm{H_1}\) and critical value but must be in context.

Scheme	Marks
Underlying (bivariate) Normal distribution	B1
	(1)

Scheme	Marks
Evidence does not support Normal distribution since mean < median or (negative) skew, oe	B1
	(1)
	(11 marks)