AS June 2018 Q3

EdexcelAS paperCurrent spec12 marksIncludes hypothesis testingCorrelation

3. The table below shows the heights cleared, in metres, for each of 6 competitors in a high jump competition.

Competitor	A	B	C	D	E	F
Height (m)	2.05	1.93	2.02	1.96	1.81	2.02

These 6 competitors also took part in a long jump competition and finished in the following order, with C jumping the furthest.

\[\mathrm{C} \qquad \mathrm{A} \qquad \mathrm{F} \qquad \mathrm{D} \qquad \mathrm{B} \qquad \mathrm{E}\]

(a) Calculate Spearman’s rank correlation coefficient for these data. (4)

(b) Stating your hypotheses clearly, test at the 5% level of significance whether or not there is a positive correlation between results in the high jump and results in the long jump. (4)

The product moment correlation coefficient between the height of the high jump and the length of the long jump for each competitor is found to be 0.678

(c) Use this value to test, at the 5% level of significance, for evidence of positive correlation between results in the high jump and results in the long jump. (2)

(d) State the condition required for the test in part (c) to be valid. (1)

(e) Explain what your conclusions in part (b) and part (c) suggest about the relationship between results in the high jump and results in the long jump. (1)

Mark scheme (a)

Scheme

Marks

Competitor	A	B	C	D	E	F
High jump ranks	1	5	2.5	4	6	2.5
Long jump ranks	2	5	1	4	6	3

M1
M1

1.1b
1.1b

\([\sum h^2 = 90.5 \quad \sum l^2 = 91 \quad \sum hl = 89]\)

Use of pmcc \(r_s = \dfrac{89 - \frac{21 \times 21}{6}}{\sqrt{\left(90.5 - \frac{21^2}{6}\right)\left(91 - \frac{21^2}{6}\right)}}\)

1.1b

\(r\) = awrt 0.899

1.1b

(4)

Notes

1^st M1 for an attempt to rank first row using tied ranks (at least 4 correct)

2^nd M1 for an attempt to rank second row (at least 4 correct)

3^rd M1 for use of pmcc with tied ranks

A1 for awrt 0.899

SC: Use of Spearman with \(\Sigma d^2\) for their ranks may score M1M1M1A0

Mark scheme (b)

Scheme	Marks	AO
\(\mathrm{H}_0\): \(\rho_s = 0 \qquad \mathrm{H}_1\): \(\rho_s \gt 0\)	B1	2.5
Critical value \(\rho_s = 0.8286\)	B1	1.1b
\(r_s = 0.899\) lies in the critical region/reject \(\mathrm{H}_0\)	M1	2.1
There is positive rank correlation between high jump and long jump results.	A1ft	2.2b
	(4)

Notes

1^st B1 both hypotheses stated in terms of \(\rho_s\) or \(\rho\)

2^nd B1 for correct critical value

M1 for comparing their ‘0.8286’ with their ‘0.899’

A1ft for a correct contextual conclusion (may ft their \(r_s\))

Mark scheme (c)

Scheme	Marks	AO
[\(\mathrm{H}_0\): \(\rho = 0 \qquad \mathrm{H}_1\): \(\rho \gt 0\)] \(0.678 \lt\) Critical value \(\rho = 0.7293\)	M1	2.1
There is no evidence of (positive) correlation (between high jump and long jump).	A1	2.2b
	(2)

Notes

M1 for comparing 0.7293 with 0.678

A1 for a correct conclusion

Scheme	Marks	AO
The test in part (c) requires the data to come from a bivariate normal distribution.	B1	2.3
	(1)

Mark scheme (e)

Scheme	Marks	AO
Although there is evidence of a positive correlation between the ranks, the data does not appear to fit a linear pattern.	B1	2.4
	(1)
	(12 marks)

Notes

B1 for comparing what each coefficient shows