S3 June 2014 Q8

EdexcelOld spec16 marksIncludes hypothesis testingCorrelation

8. The heights, in metres, and weights, in kilograms, of a random sample of 9 men are shown in the table below

Man	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)
Height (\(x\))	1.68	1.74	1.75	1.76	1.78	1.82	1.84	1.88	1.98
Weight (\(y\))	75	76	100	77	90	95	110	96	120

(a) Given that \(\mathrm{S}_{xx} = 0.0632\), \(\mathrm{S}_{yy} = 1957.5556\) and \(\mathrm{S}_{xy} = 9.3433\) calculate, to 3 decimal places, the product moment correlation coefficient between height and weight for these men. (2)

(b) Use your value of the product moment correlation coefficient to test whether or not there is evidence of a positive correlation between the height and weight of men. Use a 5% significance level. State your hypotheses clearly. (4)

Peter does not know the heights or weights of the 9 men. He is given photographs of them and asked to put them in order of increasing weight. He puts them in the order

\[A \quad C \quad E \quad B \quad G \quad D \quad I \quad F \quad H\]

(d) Use your value of Spearman’s rank correlation coefficient to test for evidence of Peter’s ability to correctly order men, by their weight, from their photographs. Use a 5% significance level and state your hypotheses clearly. (4)

Scheme	Marks
\(r = \dfrac{9.3433}{\sqrt{0.0632 \times 1957.5556}}\)	M1
= 0.840	A1
	(2)

Mark scheme (b)

Scheme	Marks
\(\mathrm{H_0}: \rho = 0 \quad \mathrm{H_1}: \rho \gt 0\)	B1
Critical value = 0.5822	B1
0.840 > 0.5822 There is evidence to reject \(\mathrm{H_0}\).	M1
There is evidence of a positive correlation between a man’s height and his weight.	A1ft
	(4)

Notes

1st B1 for both hypotheses in terms of \(\rho\), one tail \(\mathrm{H_1}\) must be compatible with their \(r\)
Hypotheses just in words e.g. “no correlation” score B0

2nd B1 for 0.5822 cao

M1 for a statement comparing ‘their r’ with ‘their cv’

A1 for a correct contextualised comment. Must mention positive correlation, be carrying out a 1-tailed test and mention height and weight.
Follow through their \(r\) and their cv (provided their \(|\text{cv}| \lt 1\) and their \(|r| \lt 1\))

Mark scheme (c)

Scheme

Marks

Man	A	B	C	D	E	F	G	H	I
Actual weight	1	2	7	3	4	5	8	6	9
Peter’s order	1	4	2	6	3	8	5	9	7
\(d^2\)	0	4	25	9	1	9	9	9	4

B1
B1

\(\sum d^2 = 70\)

M1A1

\(r_s = 1 - \dfrac{6\sum d^2}{n(n^2 - 1)}\)
\(= 1 - \dfrac{6 \times 70}{9(81 - 1)}\)

dM1

= 0.417

(6)

Notes

1st B1 for attempt to rank actual weight / Peter’s order with at least 4 correct

2nd B1 for correct rankings for both (one or both may be reversed)

1st M1 for use of \(\sum d^2\) with at least 4 values correct and attempt to add

1st A1 for 70 or 170 with reversed rankings

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

2nd A1 for awrt 0.417 or \(\dfrac{5}{12}\)

Mark scheme (d)

Scheme	Marks
\(\mathrm{H_0}: \rho = 0 \quad \mathrm{H_1}: \rho \gt 0\)	B1
Critical value 0.600	B1
(0.417 < 0.600) There is insufficient evidence to reject \(\mathrm{H_0}\).	M1
Peter does not have the ability to correctly order men, by weight, from their photograph.	A1
	(4)
	(16 marks)

Notes

1st B1 for both hypotheses in terms of \(\rho\) or \(\rho_s\) 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 0.6(00) cao
Their cv must be compatible with their \(\mathrm{H_1}\) which may be in words

M1 for statement comparing ‘their r’ with ‘their cv’

A1 for a correct contextualised comment. Must mention Peter and Men.
Follow through their \(r\) and their cv (provided their \(|\text{cv}| \lt 1\) and their \(|r_s| \lt 1\))