| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2020 |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon signed-rank test (single sample) |
| Difficulty | Moderate -0.5 Part (a) is pure recall of when to use non-parametric tests. Part (b) is a standard application of the Wilcoxon signed-rank test with clear data and hypotheses—routine for Further Statistics students with no novel problem-solving required, though the procedure itself has multiple steps. |
| Spec | 5.07a Non-parametric tests: when to use5.07b Sign test: and Wilcoxon signed-rank |
| 5.62 | 5.73 | 6.55 | 6.81 | 6.10 | 5.75 | 5.87 | 6.47 | 5.86 | 6.26 | 6.99 | 5.91 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| When the population cannot be assumed to be normally distributed | 1 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): population median is 6.00, \(H_1\): population median is greater than 6.00 | 1 | B1 |
| Calculate deviations and resulting signed ranks: 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 | M1 |
| Test statistic \(T = 8 + 7 + 5 + 3 + 4 + 1 = 28\) | 1 | A1 |
| Compare with correct critical value 17 | 1 | M1 |
| Conclusion: accept \(H_0\); insufficient evidence that the median is greater than 6.00 | 1 | A1FT |
| Total | 6 |
## Question 1(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| When the population cannot be assumed to be normally distributed | 1 | B1 | Both hypotheses stated |
---
## Question 1(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: population median is 6.00, $H_1$: population median is greater than 6.00 | 1 | B1 | Both hypotheses stated |
| Calculate deviations and resulting signed ranks: 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 | M1 | |
| Test statistic $T = 8 + 7 + 5 + 3 + 4 + 1 = 28$ | 1 | A1 | |
| Compare with correct critical value 17 | 1 | M1 | |
| Conclusion: accept $H_0$; 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 $T$ |
| **Total** | **6** | | |
---1
\begin{enumerate}[label=(\alph*)]
\item 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.
\begin{center}
\begin{tabular}{ l l l l l l l l l l l l }
5.62 & 5.73 & 6.55 & 6.81 & 6.10 & 5.75 & 5.87 & 6.47 & 5.86 & 6.26 & 6.99 & 5.91 \\
\end{tabular}
\end{center}
\item 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.\\[0pt]
[6]
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q1 [7]}}