S3 June 2013 Q2

EdexcelOld spec8 marksIncludes hypothesis testingCorrelation

2. The table below shows the number of students per member of staff and the student satisfaction scores for 7 universities.

University	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
Number of students per member of staff	14.2	13.1	13.3	11.7	10.5	15.9	10.8
Student satisfaction score	4.1	4.2	3.8	4.0	3.9	4.3	3.7

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

(b) Stating your hypotheses clearly test, at the 5% level of significance, whether or not there is evidence of a correlation between the number of students per member of staff and the student satisfaction score. (3)

Mark scheme (a)

Scheme

Marks

Uni	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
Staff-Stu	2	4	3	5	7	1	6
Satisfaction	3	2	6	4	5	1	7
[ \(d\) ]	−1	2	−3	1	2	0	−1
\(d^2\)	1	4	9	1	4	0	1	20

M1A1A1

\(r_s = 1 - \dfrac{6 \times 20}{7(49 - 1)} = 0.642857\ldots\) ( accept \(\dfrac{9}{14}\)) (awrt 0.643)

dM1A1

(5)

Notes

1st M1 for an attempt to rank the staff-students ratio or satisfaction ( at least 4 correct)

1st A1 for correct rankings for both (one or both may be reversed)

2nd A1 for \(\sum d^2 = 20\) or correct \(d^2\) row (NB \(\sum d^2 = 92\) for one set of reversed ranks)

2nd dM1 for use of the correct formula, follow through their \(\sum d^2\) (Dependent on 1^st M1)
If answer is not correct, a correct expression is required.

3rd A1 If \(\sum d^2 = 20\) for awrt 0.643 or if \(\sum d^2 = 92\) for awrt – 0.643 (accept \(\pm\tfrac{9}{14}\))

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : \rho = 0\) \(\mathrm{H}_1 : \rho \ne 0\) (\(\rho \gt 0\))	B1
Critical value is \(\pm 0.7857\) (\(\pm 0.7143\) for a one tailed test)	B1
0.643<cv so insufficient evidence to reject \(\mathrm{H}_0\) There is insufficient evidence to suggest a (positive) correlation between staff-student ratio and satisfaction.	B1ft
	(3)
	(8 marks)

Notes

1st B1 for both hypotheses in terms of \(\rho\), one tail \(\mathrm{H}_1\) must be compatible with their ranking
Hypotheses just in words e.g. “no correlation” score B0

2nd B1 for cv of 0.7857 or 0.7143 for one-tailed test (accept \(\pm\))
Their cv must be compatible with their \(\mathrm{H}_1\) which may be in words
If hypotheses are the wrong way around this must be B0 but 3^rd B1 is possible.

3rd B1ft for a correct contextualised comment. Must mention “ratio” or “no. of students per member of staff” and “satisfaction”
Follow through their \(r_s\) and their cv (provided it is |cv| <1)
Don’t insist on the word “positive” for a one-tailed test
Use of “association” is B0
Independent of 1^st B1 so if \(|r_s| \gt |\text{cv}|\) must say there is sufficient evidence of …….(o.e.)
and if \(|r_s| \lt |\text{cv}|\) must say insufficient evidence of … (o.e.) regardless of their hypotheses
Contradictory statements score B0
(This mark is just testing interpretation of comparison of their \(r_s\) and their cv)