| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2020 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Justifying choice of test |
| Difficulty | Moderate -0.3 Part (a) requires recalling standard conditions for Wilcoxon rank-sum test (independent samples, ordinal data, no normality assumption) - straightforward bookwork. Part (b) involves ranking 24 values, summing ranks, and comparing to critical values - computational but routine application of a standard procedure with no conceptual challenges beyond careful arithmetic. |
| Spec | 5.07a Non-parametric tests: when to use5.07b Sign test: and Wilcoxon signed-rank5.07d Paired vs two-sample: selection |
| Group \(A\) | 12.3 | 11.8 | 12.1 | 13.2 | 11.1 | 10.6 | 13.8 | 12.0 | 12.2 | 12.4 | 13.5 | 13.9 |
| Group \(B\) | 11.7 | 10.8 | 10.9 | 11.3 | 11.2 | 12.6 | 11.0 | 10.5 | 11.9 | 12.5 | 10.7 | 11.6 |
| Answer | Marks |
|---|---|
| Difference of location test for populations not known to be normal | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Ranking table completed correctly (two columns: values with ranks 10–24 for x-group, values with ranks 1–20 for y-group as shown) | M1 | Method for ranking all 24 values |
| Total ranks: 109 or 191 | A1 | Either correct rank sum |
| \(H_0: m_x = m_y \quad H_1: m_x > m_y\) | B1 | Both hypotheses correct in terms of medians |
| Use normal approximation with attempts at mean and variance | M1 | |
| mean \(= 150\), variance \(= 300\) | A1 | Both correct |
| \(\dfrac{109.5 - 150}{\sqrt{300}} = -2.338\) | M1A1 | Continuity correction \(\pm 0.5\) applied; correct z-value |
| Probability \(= 0.0097\) | A1 | |
| This is less than \(0.05\), so reject \(H_0\) | M1 | Correct comparison and conclusion |
| There is evidence of an effect on growth | A1 | Conclusion in context |
| Total | 10 |
## Question 6:
**Part 6(a):**
Difference of location test for populations not known to be normal | B1 |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Ranking table completed correctly (two columns: values with ranks 10–24 for x-group, values with ranks 1–20 for y-group as shown) | M1 | Method for ranking all 24 values |
| Total ranks: 109 or 191 | A1 | Either correct rank sum |
| $H_0: m_x = m_y \quad H_1: m_x > m_y$ | B1 | Both hypotheses correct in terms of medians |
| Use normal approximation with attempts at mean and variance | M1 | |
| mean $= 150$, variance $= 300$ | A1 | Both correct |
| $\dfrac{109.5 - 150}{\sqrt{300}} = -2.338$ | M1A1 | Continuity correction $\pm 0.5$ applied; correct z-value |
| Probability $= 0.0097$ | A1 | |
| This is less than $0.05$, so reject $H_0$ | M1 | Correct comparison and conclusion |
| There is evidence of an effect on growth | A1 | Conclusion in context |
| **Total** | **10** | |6 A biologist is studying the effect of nutrients on the heights to which plants grow. A random sample of 24 similar young plants is divided into two equal groups $A$ and $B$. The plants in group $A$ are fed with nutrients and water and the plants in group $B$ are given only water. After four weeks, the height, in cm, of each plant is measured and the results are as follows.
\begin{center}
\begin{tabular}{ | l | l | l | l | l | l | l | l | l | l | l | l | l | }
\hline
Group $A$ & 12.3 & 11.8 & 12.1 & 13.2 & 11.1 & 10.6 & 13.8 & 12.0 & 12.2 & 12.4 & 13.5 & 13.9 \\
\hline
Group $B$ & 11.7 & 10.8 & 10.9 & 11.3 & 11.2 & 12.6 & 11.0 & 10.5 & 11.9 & 12.5 & 10.7 & 11.6 \\
\hline
\end{tabular}
\end{center}
The biologist decides to carry out a test at the $5 \%$ significance level to test whether the nutrients have resulted in an increase in growth.
\begin{enumerate}[label=(\alph*)]
\item She carries out a Wilcoxon rank-sum test. Give a reason why this is an appropriate choice of test.
\item Carry out the Wilcoxon rank-sum test for these results.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q6 [11]}}