S3 June 2013 (R) Q3

EdexcelOld spec13 marksIncludes hypothesis testingCorrelation

3. The table below shows the population and the number of council employees for different towns and villages.

Town or village	Population	Number of council employees
\(A\)	211	10
\(B\)	356	2
\(C\)	1047	12
\(D\)	2463	21
\(E\)	4892	16
\(F\)	6479	25
\(G\)	6571	67
\(H\)	6573	45
\(I\)	9845	48
\(J\)	14 784	34

(a) Find, to 3 decimal places, Spearman’s rank correlation coefficient between the population and the number of council employees. (5)

(b) Use your value of Spearman’s rank correlation coefficient to test for evidence of a positive correlation between the population and the number of council employees. Use a 2.5% significance level. State your hypotheses clearly. (4)

It is suggested that a product moment correlation coefficient would be a more suitable calculation in this case. The product moment correlation coefficient for these data is 0.627 to 3 decimal places.

(c) Use the value of the product moment correlation coefficient to test for evidence of a positive correlation between the population and the number of council employees. Use a 2.5% significance level. (2)

(d) Interpret and comment on your results from part (b) and part (c). (2)

Mark scheme (a)

Scheme

Marks

Town	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Pop	1	2	3	4	5	6	7	8	9	10
Empl	2	1	3	5	4	6	10	8	9	7
\(\|d\|\)	1	1	0	1	1	0	3	0	0	3
\(d^2\)	1	1	0	1	1	0	9	0	0	9

\(\sum d^2 = 22\)

M1A1

\(r_s = 1 - \dfrac{6 \times 22}{10 \times 99}\)

dM1

\(= \dfrac{143}{165} = 0.86\dot{6}\) awrt 0.867

(5)

Notes

1st M1 for an attempt to rank no of employees against the populations

2nd M1 for attempting \(\sum d^2\) (must be using ranks) ft their ranks

1st A1 for 22

3rd dM1 dep on 1st M1 for use of the correct formula with their \(\sum d^2\).
If ans. is not correct an expr’ is required.

Mark scheme (b)

Scheme	Marks
\(\mathrm{H_0}: \rho = 0\ ;\ \mathrm{H_1}: \rho \gt 0\)	B1
CV = 0.6485	B1
in critical region / significant/ reject \(\mathrm{H_0}\)	M1
evidence of positive correlation between population and no. of employees	A1
	(4)

Notes

1st B1 for both hypotheses in terms of \(\rho\), \(\mathrm{H_1}\) must be one tail and compatible with their ranking

M1 for a correct statement relating their \(r_s\) (\(|r_s| \lt 1\)) with their cv but cv must be such that \(|\text{cv}| \lt 1\)

A1 for a correct contextualised comment that is rejecting \(\mathrm{H_0}\)
Must mention “population” and “no. of employees” and “positive correlation”.
Follow through their \(r_s\) and their cv (provided it is \(|\text{cv}| \lt 1\)) Use of “association” is A0

Mark scheme (c)

Scheme	Marks
CV = 0.6319	B1
[not in critical region / not significant/ do not reject \(\mathrm{H_0}\)] No evidence of positive correlation	B1
	(2)

Notes

1st B1 for 0.6319

2nd B1 does not require context just no positive correlation mentioned

Mark scheme (d)

Scheme	Marks
No evidence to suggest that as pop’ increased the no. of employees increased linearly.	B1
Villages ranked highly for pop’ were also ranked highly for the no. of employees.	B1
	(2)
	(13 marks)

Notes

1st B1 for a comment relating to pmcc
(i) no linear relationship or (ii) pmcc requires (joint) normal distribution

2nd B1 for a second comment relating to Spearman’s
(i) there is a (non-linear) relationship between ranks.. or (ii) data not (joint) normal so Spearman’s is better

ALT

Scheme	Marks
Alternate for part (d) if different conclusions in part (b) and part (c) Data probably not (bivariate) normal therefore Spearman's coefficient is more suitable than the product moment correlation coefficient.