S3 June 2009 Q3

EdexcelOld spec11 marksIncludes hypothesis testingCorrelation

3. A doctor is interested in the relationship between a person’s Body Mass Index (BMI) and their level of fitness. She believes that a lower BMI leads to a greater level of fitness. She randomly selects 10 female 18 year-olds and calculates each individual’s BMI. The females then run a race and the doctor records their finishing positions. The results are shown in the table.

Individual	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
BMI	17.4	21.4	18.9	24.4	19.4	20.1	22.6	18.4	25.8	28.1
Finishing position	3	5	1	9	6	4	10	2	7	8

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

(b) Stating your hypotheses clearly and using a one tailed test with a 5% level of significance, interpret your rank correlation coefficient. (5)

(c) Give a reason to support the use of the rank correlation coefficient rather than the product moment correlation coefficient with these data. (1)

Mark scheme (a)

Scheme

Marks

	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
BMI	1	6	3	8	4	5	7	2	9	10
or	10	5	8	3	7	6	4	9	2	1
Finishing position	3	5	1	9	6	4	10	2	7	8
\(d^2\)	4	1	4	1	4	1	9	0	4	4

\(\sum d^2 = 32\) (298)

\(r_s = 1 - \dfrac{6 \times 32}{10 \times 99}\)

M1 A1ft

\(= 0.80606\ldots\) \((-0.80606)\) accept \(\pm\dfrac{133}{165}\) awrt \(\pm\) 0.806

(5)

Notes

1st M1 for attempt to rank BMI scores

2nd M1 for attempt at \(\sum d^2\) (must be using ranks)

3rd M1 for use of the correct formula with their \(\sum d^2\). If answer is not correct an expression is required.

1st A1ft for a correct expression. ft their \(\sum d^2\) but only if all 3 Ms are scored

2nd A1 awrt \(\pm\) 0.806 (but sign must be compatible with their \(\sum d^2\))

No ranking can score 3^rd M1 only

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : \rho = 0, \ \mathrm{H}_1 : \rho \gt 0\),	B1 B1
Critical value is \((\pm)0.5636\)	B1
(0.806 > 0.5636 therefore) in critical region/ reject \(\mathrm{H}_0\)	M1
The lower the BMI the higher the position in the race./ support for doctors belief	A1ft
	(5)

Notes

2nd B1 for \(\rho \gt 0\) (or <0 but must be one tail and consistent with their ranking)

3rd B1 for critical value that is compatible with their \(\mathrm{H}_1\). If one-tail must be \(\pm\) 0.5636 if two-tail must be \(\pm\) 0.6485 [Condone wrong sign]

M1 for a correct statement relating their \(r_s\) with their cv. e.g. “reject \(\mathrm{H}_0\)”, “in critical region”, “significant result”
May be implied by a correct comment

A1ft for correct comment in context. Must mention low/high BMI and race/fitness or doctor’s belief. Comment should be one-tailed. Allow positive correlation between… but NOT …positive relationship…

No \(\mathrm{H}_1\) assume one-tail for 3^rd B1

Mark scheme (c)

Scheme	Marks
The position is already ranked OR Position is not Normally distributed	B1
	(1)
	(11 marks)

Notes

B1 for a correct and relevant comment either based on the fact that the data was originally partially ordered or on the underlying normal assumption
“Quicker” or “easier” score B0