S3 June 2010 Q4

EdexcelOld spec10 marksIncludes hypothesis testingCorrelation

4. A researcher claims that, at a river bend, the water gradually gets deeper as the distance from the inner bank increases. He measures the distance from the inner bank, \(b\) cm, and the depth of a river, \(s\) cm, at seven positions. The results are shown in the table below.

Position	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)
Distance from inner bank \(b\) cm	100	200	300	400	500	600	700
Depth \(s\) cm	60	75	85	76	110	120	104

(a) Calculate Spearman’s rank correlation coefficient between \(b\) and \(s\). (6)

(b) Stating your hypotheses clearly, test whether or not the data provides support for the researcher’s claim. Use a 1% level of significance. (4)

Mark scheme (a)

Scheme

Marks

Distance rank	1	2	3	4	5	6	7
Depth rank	1	2	4	3	6	7	5

\(\|d\|\)	0	0	1	1	1	1	2
\(d^2\)	0	0	1	1	1	1	4

\(\sum d^2 = 8\)

M1A1

\(r_s = 1 - \dfrac{6 \times 8}{7 \times 48}\)

\(= \dfrac{6}{7} = 0.\dot{8}5714\dot{2}\) awrt 0.857

(6)

Notes

1st M1 for an attempt to rank the depths against the distances

2nd M1 for attempting \(d\) for their ranks. Must be using ranks.

3rd M1 for attempting \(\sum d^2\) (must be using ranks)

1st A1 for sum of 8 (or 104 for reverse ranking)

4th M1 for use of the correct formula with their \(\sum d^2\). If answer is not correct an expression is required.

2nd A1 for awrt (\(\pm\)) 0.857. Sign should correspond to ranking (so use of 104 should get -0.857)

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : \rho = 0, \mathrm{H}_1 : \rho \gt 0\)	B1
Critical value at 1% level is 0.8929	B1
\(r_s \lt 0.8929\) so not significant evidence to reject \(\mathrm{H}_0\),	M1
The researcher’s claim is not correct (at 1% level). or insufficient evidence for researcher’s claim or there is insufficient evidence that water gets deeper further from inner bank. or no (positive) correlation between depth of water and distance from inner bank	A1ft
	(4)
	(10 marks)

Notes

1st B1 for both hypotheses in terms of \(\rho\), \(\mathrm{H}_1\) must be one tail and compatible with their ranking

2nd B1 for cv of 0.8929 (accept \(\pm\))

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

A1ft for a correct contextualised comment. Must mention “researcher” and “claim” or “distance (from bank)” and “depth (of water)”
Follow through their \(r_s\) and their cv (provided it is |cv| <1)
Use of “association” is A0