CAIE A-Level Further Mathematics 4.4 Non-Parametric Tests

Practise selecting and carrying out sign, signed-rank or rank-sum tests, using population-median hypotheses, correct ranks and contextual conclusions.

Exam points

choose a sign, signed-rank or rank-sum procedure from the pairing and distribution assumptions
state population-median hypotheses and calculate the signed or combined ranks accurately
compare the test statistic or probability with its threshold and conclude in the claim's context

Covered syllabus points

4.4.1—Non-parametric tests 4.4.2—The basis of the sign test 4.4.3—Single-sample sign tests 4.4.4—Paired non-parametric tests

[Maximum number: 7]

(a)

State briefly the circumstances under which a non-parametric test of significance should be used rather than a parametric test.

The level of pollution in a river was measured at 12 randomly chosen locations. The results, in suitable units, are shown below, where higher values represent greater pollution.

Table for Question (a) — CAIE A-Level Further Math AS

[ 1 ]

(b)

Use a Wilcoxon signed-rank test to test whether the average pollution level in the river is more than 6.00. Use a 5% significance level.

[ 6 ]

$\mathrm{H}_{0}$ : population median is $6.00, \mathrm{H}_{1}$ : population median is greater than 6.00

1

B1

Both hypotheses stated

Calculate deviations and resulting signed ranks

1

M1

Devs & -0.38 & -0.27 & 0.55 & 0.81 & 0.10 & -0.25 & -0.13 & 0.47 & -0.14 & 0.26 & 0.99 & -0.09

Rank & -8 & -7 & 10 & 11 & 2 & -5 & -3 & 9 & -4 & 6 & 12 & -1

1

A1

Test statistic T=8+7+5+3+4+1=28

1

A1

Compare with correct critical value 17

1

M1

Conclusion: accept $\mathrm{H}_{0}$ ; insufficient evidence that the median is greater than
6.00

1

A1FT

Conclusion to be stated in context, not just 'not
significant'; follow through their value of T

6

[Maximum number: 8]

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 Non-parametric tests question 2 — CAIE A-Level Further Math AS

(a)

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

[ 7 ]

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

(b)

State an assumption that you have made in carrying out the test in part (a).

[ 1 ]

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Covered syllabus points

4.4 Non-parametric tests question 1

Question (a)

Question (b)

4.4 Non-parametric tests question 2

Question (a)

Question (b)