AS June 2023 Q1

EdexcelAS paperCurrent spec10 marksIncludes hypothesis testingCorrelation

1. Every applicant for a job at Donala is given three different tasks, \(P\), \(Q\) and \(R\).
For each task the applicant is awarded a score.
The scores awarded to 9 of the applicants, for the tasks \(P\) and \(Q\), are given below.

Applicant	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)
Task P	19	16	16	12	8	17	12	12	5
Task Q	17	11	14	7	6	18	15	11	10

(a) Calculate Spearman’s rank correlation coefficient for the scores awarded for the tasks \(P\) and \(Q\). (4)

(b) Test, at the 1% level of significance, whether or not there is evidence for a positive correlation between the ranks of scores for tasks \(P\) and \(Q\).
You should state your hypotheses and critical value clearly. (4)

The Spearman’s rank correlation coefficient for \(P\) and \(R\) is 0.290 and for \(Q\) and \(R\) is 0.795

The manager of Donala wishes to reduce the number of tasks given to job applicants from three to two.

Mark scheme (a)

Scheme

Marks

Applicant Task	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)
\(P\)	1	3.5	3.5	6	8	2	6	6	9
\(Q\)	2	5.5	4	8	9	1	3	5.5	7

M1
M1

1.1b
1.1b

\(\sum p^2 = 282.5 \qquad \sum q^2 = 284.5 \qquad \sum pq = 271.25\)

\(r_s = \dfrac{\text{“}271.25\text{”} - \dfrac{45 \times 45}{9}}{\sqrt{\left(\text{“}282.5\text{”} - \dfrac{45^2}{9}\right)\left(\text{“}284.5\text{”} - \dfrac{45^2}{9}\right)}}\) Must be using ranks

1.1b

\(r_s = 0.7907\ldots\) awrt 0.791

1.1b

(4)

Notes

M1: For an attempt to rank at least one row (at least 4 correct) [May have no tied ranks]

M1: For an attempt to rank both rows using tied ranks (at least 4 correct)

M1: for those who have incorrect rankings but show some working of
\(r_s = \dfrac{\text{“}271.25\text{”} - \dfrac{45 \times 45}{9}}{\sqrt{\left(\text{“}282.5\text{”} - \dfrac{45^2}{9}\right)\left(\text{“}284.5\text{”} - \dfrac{45^2}{9}\right)}}\) May be implied by a correct answer

A1: awrt 0.791 (NB 0.796 is M1M1 M1A0)

Alternative

ALT For use of \(\sum d^2 = 1^2 + 2^2 + 0.5^2 + 2^2 + 1^2 + 1^2 + 3^2 + 0.5^2 + 2^2\ [= 24.5]\) - allow use of "their \(d\)" and
\(r_s = 1 - \dfrac{6 \times \text{“}24.5\text{”}}{9 \times 80}\ (= 0.7958\ldots)\) they can get M1M1M1A0 (But dep on some attempt to rank)

Mark scheme (b)

Scheme	Marks	AO
\(\mathrm{H}_0 : \rho = 0 \quad \mathrm{H}_1 : \rho \gt 0\)	B1	2.5
Critical Value \(\rho = 0.7833\)	B1	1.1b
\(r_s = 0.7907\) lies in the critical region/reject \(\mathrm{H}_0\)/significant	M1	2.1
There is evidence of a positive correlation between the ranks of scores for tasks \(P\) and \(Q\)	A1ft	2.2b
	(4)

Notes

B1: Both hypotheses stated in terms of \(\rho\) or \(\rho_S\)

B1: for the correct critical value of 0.7833 (use of pmcc 0.7498 is B0)

M1: for comparing their 0.791 with their 0.7833 (could have 0.7498) and making a correct statement

A1ft: For a correct ft contextual conclusion with no contradictions seen.
Must mention “scores” and either “task” or “\(P\) and \(Q\)”

Mark scheme (c)

Scheme	Marks	AO
\(P\) and \(R\) are not in agreement so they will give different information about the applicants (oe) or \(P\) and \(Q\) and \(Q\) and \(R\) are in some agreement so they will give similar information about the applicants (oe)	M1	2.4
\(P\) and \(R\)	A1	2.2b
	(2)
	(10 marks)

Notes

M1: For a correct explanation to support their answer. Must have idea of “different” or “similar” tasks

A1: for a correct deduction from the information i.e. choosing \(P\) and \(R\) only