CAIE A-Level Further Mathematics 4.4.4 Paired Non-Parametric Tests

Practise applying matched-pairs sign or signed-rank tests and independent rank-sum tests, including how corrected observations change ranks and conclusions.

[Maximum number: 7]

A large number of students took two test papers in mathematics. The teacher believes that the marks obtained in Paper 1 will be higher than the marks obtained in Paper 2. She chooses a random sample of 9 students and compares their marks. The marks are shown in the table.

Table for Question 4.4.4—Paired non-parametric tests question 1 — CAIE A-Level Further Math A2

Carry out a Wilcoxon matched-pairs signed-rank test, at the 5% significance level, to test whether the data supports the teacher's belief.

5 & 7 & -6 & 1 & -4 & 2 & 8 & 26 & -10

4 & 6 & -5 & 1 & -3 & 2 & 7 & 9 & -8

M1

Differences, allow at most 2 errors.

A1

Correct rank order, ignore signs.

[ P=29 ] Q=16

A1

Condone P not excluded.

$\mathrm{H}_{0}$ : population medians equal or $m_{1}=m_{2}$ $\mathrm{H}_{1}$ : population median X> population median Y or $m_{1}>m_{2}$

B1

'Population' required. Accept use of m, not $\mu$ .

Do not accept 'difference between population medians >0 ' without X, Y OE specified.

Critical value =8

B1

16>8 and accept $\mathrm{H}_{\mathrm{o}}$ / not significant

M1

Compare their ' 16 ' with their ' 8 ' and conclusion.
Their '16' must be less than 23. Ignore their hypotheses.
Condone 'reject $\mathrm{H}_{1}$ '.

Insufficient evidence to support teacher's belief
or
insufficient evidence that the marks/median in Paper 1 are/is
higher than the marks/median in Paper 2

A1

Correct conclusion in context, following correct work, level of
uncertainty in language (not 'prove', not 'there is no evidence'),
no contradictions.
e.g. Proves that the teacher is incorrect scores A0. A0 if
hypotheses wrong way round.

7

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4.4.4—Paired non-parametric tests question 1