AS June 2024 Q2

EdexcelAS paperCurrent spec7 marksIncludes hypothesis testingCorrelation

2. A random sample of size \(n = 8\) of paired data is taken from a population. The data are plotted below.

Scatter diagram of 8 points with x and y axes from 0 to 40, showing y generally decreasing as x increases

Test, at the 1% level of significance, whether or not there is evidence of a negative rank correlation between the two variables.

You should state your hypotheses and critical value and show your working clearly. (7)

Mark scheme

Scheme

Marks

\(\mathrm{H}_0 : \rho_s = 0 \qquad \mathrm{H}_1 : \rho_s \lt 0\)

2.5

\(x\)	1	2	3	4	5	6	7	8
\(y\)	7	8	5	6	3	4	2	1
\(d^2\)	36	36	4	4	4	4	25	49

\(\sum d^2 = 162\)

M1
dM1

3.1a
1.1b

\(r_s = 1 - \dfrac{6 \times \text{“}\textit{their} \sum d^2\text{”}}{8(8^2 - 1)}\)

1.1b

\(r_s = -0.9285\ldots\) awrt –0.929

1.1b

Critical value –0.8333

1.1b

(Reject \(\mathrm{H}_0\)) there is sufficient evidence to suggest the data shows negative rank correlation.

A1ft

2.2b

(7)

(7 marks)

Notes

1^st B1: Both hypotheses correct in terms of \(\rho_s\) or \(\rho\)

1^st M1: for an attempt to rank both rows (at least 4 correct in one row or 2 pairs).
Condone reverse rankings.

\(x\)	1	2	3	4	5	6	7	8
\(y\)	2	1	4	3	6	5	7	8
\(d^2\)	1	1	1	1	1	1	0	0

\(\sum d^2 = 6\)

2^nd dM1: dep on 1^st M1 for attempt at \(d\) or \(d^2\) row for their rankings.

3^rd M1: for use of \(1 - \dfrac{6 \times \text{“}162\text{”}}{8(8^2 - 1)}\) Allow independently of 1^st M1 and 2^nd M1

1^st A1: awrt –0.929 allow \(-\dfrac{13}{14}\) or –0.928 following a correct expression
Allow for + 0.929 etc following correct reverse ranking using \(\sum d^2 = 6\)

2^nd B1: correct critical value (allow \(\pm\) 0.8333). 4dp here but allow 3 in 2^nd A1

2^nd A1ft: dependent upon all M marks and \(|r_s| = 0.929\) or \(|\text{cv}| = 0.833\) (or better)
No incorrect statements e.g. “accept \(\mathrm{H}_0\) there is evidence of negative …”
If comparison is seen it must be correct:
e.g. \(-0.929 \gt -0.833\) or \(-0.929 \lt 0.833\) are both A0
Allow “insufficient evidence of negative rank correlation” provided it follows from their \(r_s\) and their cv (at least one of which must be correct).