CAIE A-Level Further Mathematics 4.4.3 Single-Sample Sign and Signed-Rank Tests

Practise testing a population median with signs or signed ranks, excluding zero differences, selecting the smaller rank sum and using exact or normal critical values.

Exam points

state hypotheses about the population median and subtract the hypothesised value consistently
rank absolute non-zero differences, restore signs and use the required positive or smaller rank sum
compare with the exact table or an appropriate normal approximation before concluding in context

[Maximum number: 6]

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.

$\mathrm{H}_{0}$ : 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

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4.4.3—Single-sample sign tests question 1