S3 June 2005 Q4

EdexcelOld spec13 marksIncludes hypothesis testingCorrelation

4. Over a period of time, researchers took 10 blood samples from one patient with a blood disease. For each sample, they measured the levels of serum magnesium, \(s\) mg/dl, in the blood and the corresponding level of the disease protein, \(d\) mg/dl. The results are shown in the table.

\(s\)	1.2	1.9	3.2	3.9	2.5	4.5	5.7	4.0	1.1	5.9
\(d\)	3.8	7.0	11.0	12.0	9.0	12.0	13.5	12.2	2.0	13.9

[Use \(\sum s^2 = 141.51\), \(\sum d^2 = 1081.74\) and \(\sum sd = 386.32\)]

(a) Draw a scatter diagram to represent these data. (3)

(b) State what is measured by the product moment correlation coefficient. (1)

(d) Calculate the value of the product moment correlation coefficient \(r\) between \(s\) and \(d\). (2)

(e) Stating your hypotheses clearly, test, at the 1% significance level, whether or not the correlation coefficient is greater than zero. (3)

(f) With reference to your scatter diagram, comment on your result in part (e). (1)

Scheme	Marks
NB No graph paper \(\Rightarrow\) 0/3	B1 B2
	(3)

Scheme	Marks
Linear association between \(s\) and \(d\)	B1
	(1)

Scheme	Marks
\(S_{ss} = 141.51 - \dfrac{33.9^2}{10} = \underline{26.589}\); \(S_{dd} = \underline{152.444}\); \(S_{sd} = \underline{59.524}\)	B1; B1; B1
	(3)

Scheme	Marks
\(r = \dfrac{59.524}{\sqrt{152.444 \times 26.589}}\)	M1
\(= \underline{0.93494\ldots}\) AWRT 0.935	A1
	(2)

Scheme	Marks
\(\mathrm{H}_0: \rho = 0\); \(\mathrm{H}_1: \rho \gt 0\)	B1
Critical value at 1% = 0.7155	B1
Reject \(\mathrm{H}_0\); levels of serum & disease are positively correlated	B1
	(3)

Notes