S3 June 2018 Q1

EdexcelOld spec13 marksIncludes hypothesis testingCorrelation

1. Phil measures the concentration of a radioactive element, \(c\), and the amount of dissolved solids, \(a\), of 8 random samples of groundwater. His results are shown in the table below.

Sample	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
\(c\)	625	700	650	645	720	600	825	665
\(a\)	1.28	1.30	1.00	1.20	1.55	1.15	1.40	1.45

Given that

\[\mathrm{S}_{cc} = 34\,787.5 \qquad \mathrm{S}_{aa} = 0.217\,287\,5 \qquad \mathrm{S}_{ca} = 47.7625\]

(a) calculate, to 3 decimal places, the product moment correlation coefficient between the concentration of the radioactive element and the amount of dissolved solids for these groundwater samples. (1)

(b) Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the concentration of this radioactive element and the amount of dissolved solids in groundwater. Use a 5% significance level. State your hypotheses clearly. (3)

(c) Calculate, to 3 decimal places, Spearman’s rank correlation coefficient between the concentration of the radioactive element and the amount of dissolved solids. (5)

(d) Use your value of Spearman’s rank correlation coefficient to test for evidence of a positive correlation between the concentration of the radioactive element and the amount of dissolved solids. Use a 5% significance level. State your hypotheses clearly. (3)

(e) Using your conclusions in part (b) and part (d), comment on the possible relationship between these variables. (1)

Scheme	Marks
\(r = \dfrac{\mathrm{S}_{ca}}{\sqrt{\mathrm{S}_{cc}\mathrm{S}_{aa}}} = \dfrac{47.7625}{\sqrt{34787.5 \times 0.217287}} = 0.549361\ldots\)	B1
	(1)

Mark scheme (b)

Scheme	Marks
\(\mathrm{H}_0 : \rho = 0,\ \mathrm{H}_1 : \rho \gt 0\)	B1
(0.549 <)0.6215 (Not significant Insufficient evidence to reject \(\mathrm{H}_0\))	B1
Insufficient evidence of a positive correlation between the concentration of a radioactive element and the amount of dissolved solids in groundwater.	B1ft
	(3)

Notes

1st B1 Both correct. Require population parameter \(\rho\) and one tailed test.

2nd B1 cv 0.6215

3rd B1 Context required. Must mention concentration of a radioactive element and amount of dissolved solids

Mark scheme (c)

Scheme

Marks

Sample	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)
\(c\)	2	6	4	3	7	1	8	5
\(a\)	4	5	1	3	8	2	6	7

\(d\)	-2	1	3	0	-1	-1	2	-2
\(d^2\)	4	1	9	0	1	1	4	4	24

M1A1

Note Reverse ranks \(\sum d^2 = 144\)

\(r_s = 1 - \dfrac{6\sum d^2}{n(n^2 - 1)} = 1 - \dfrac{6 \times 24}{8(64 - 1)} = 0.71428\ldots\)

M1A1

(5)

Notes

1st M1 for an attempt to rank the concentration of a radioactive element and the amount of dissolved solids with at least 4 correct for each variable. Allow reverse ranks.

2nd M1 for attempt at \(d^2\) row

1st A1 all correct

3rd M1 for use of the correct formula and an attempt to rank, follow through their \(\sum d^2\) if clearly stated
If answer is not correct, a correct expression is required.

A1 awrt 0.714

Mark scheme (d)

Scheme	Marks
\(\mathrm{H}_0 : \rho_s = 0,\ \mathrm{H}_1 : \rho_s \gt 0\)	B1
(0.714 >)0.6429 (Significant. Reject \(\mathrm{H}_0\))	B1
Evidence of a positive correlation between the concentration of a radioactive element and the amount of dissolved solids in groundwater.	B1ft
	(3)

Notes

1st B1 for both hypotheses in terms of \(\rho\), one tail \(\mathrm{H}_1\). Allow use of \(\rho_s\).
Alternative hypothesis compatible with their ranking.

2nd B1 for cv of 0.6429

3rd B1ft for a correct contextualised comment. Must mention concentration of a radioactive element and the amount of dissolved solids. 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.

Mark scheme (e)

Scheme	Marks
Results of tests suggest (monotonic) non-linear relationship or assumptions for PMCC breached i.e. not (joint) normal.	B1
	(1)
	(13 marks)

Notes

B1 for ‘non-linear’ oe, or ‘not normal’