| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2020 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon signed-rank test (single sample) |
| Difficulty | Standard +0.3 This is a straightforward application of the Wilcoxon signed-rank test with clear steps: calculate differences from the hypothesized median, rank absolute differences, sum ranks for negative differences, and compare to critical value. Part (b) requires simple recall of test properties. While it's a Further Maths topic, the execution is mechanical with no conceptual challenges or novel problem-solving required. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: m = 200\), \(H_1: m < 200\) | B1 | Allow 'median' in words, allow \(m\) not defined |
| \(10, -14, -12, 8, -16, -9, 15, -2, -4\) | M1 | Signed differences (at most 3 errors) |
| Ranks: \(5, -7, -6, 3, -9, -4, 8, -1, -2\) | A1 | Award for correct rank order, ignore signs |
| Sum ranks \(T = 16\) | A1 | CWO |
| Critical value 8 and compare \(16 > 8\) | M1 | Compare their \(T\) with 8 |
| Accept \(H_0\); insufficient evidence to support scientist's belief | A1 | In context, all correct, except possibly hypotheses. Level of uncertainty in language used. No contradictions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Magnitude of differences from median are taken into account | B1 | Must mention magnitude and differences |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: m = 200$, $H_1: m < 200$ | B1 | Allow 'median' in words, allow $m$ not defined |
| $10, -14, -12, 8, -16, -9, 15, -2, -4$ | M1 | Signed differences (at most 3 errors) |
| Ranks: $5, -7, -6, 3, -9, -4, 8, -1, -2$ | A1 | Award for correct rank order, ignore signs |
| Sum ranks $T = 16$ | A1 | CWO |
| Critical value 8 and compare $16 > 8$ | M1 | Compare their $T$ with 8 |
| Accept $H_0$; insufficient evidence to support scientist's belief | A1 | In context, all correct, except possibly hypotheses. Level of uncertainty in language used. No contradictions |
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Magnitude of differences from median are taken into account | B1 | Must mention magnitude and differences |2 Metal rods produced by a certain factory are claimed to have a median breaking strength of 200 tonnes. For a random sample of 9 rods, the breaking strengths, measured in tonnes, were as follows.
$$\begin{array} { l l l l l l l l l }
210 & 186 & 188 & 208 & 184 & 191 & 215 & 198 & 196
\end{array}$$
A scientist believes that the median breaking strength of metal rods produced by this factory is less than 200 tonnes.
\begin{enumerate}[label=(\alph*)]
\item Use a Wilcoxon signed-rank test, at the $5 \%$ significance level, to test whether there is evidence to support the scientist's belief.
\item Give a reason why a Wilcoxon signed-rank test is preferable to a sign test, when both are valid.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q2 [7]}}