| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2023 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon matched-pairs signed-rank test |
| Difficulty | Standard +0.3 This is a straightforward application of the Wilcoxon matched-pairs signed-rank test with clear data and standard procedure. Students must calculate differences, rank them, sum ranks, and compare to critical values. While it requires careful execution of multiple steps, it's a routine textbook exercise with no conceptual challenges or novel insights required. The assumption in part (b) is standard recall. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07c Single-sample tests |
| Student | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) |
| Paper 1 | 46 | 73 | 55 | 64 | 86 | 42 | 66 | 68 | 60 |
| Paper 2 | 41 | 66 | 61 | 63 | 90 | 40 | 58 | 42 | 70 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Differences table: 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. |
| Correct rank order, ignore signs | A1 | |
| \([P = 29]\ Q = 16\) | A1 | Condone \(P\) not excluded. |
| \(H_0\): population medians equal or \(m_1 = m_2\); \(H_1\): 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 \(H_0\) / not significant | M1 | Compare their '16' with their '8' and conclusion. Their '16' must be less than 23. Ignore their hypotheses. Condone 'reject \(H_1\)'. |
| 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. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| The population differences are symmetrical (about the median difference) | B1 | Words in bold, or their equivalent, are required. |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Differences table: 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. |
| Correct rank order, ignore signs | A1 | |
| $[P = 29]\ Q = 16$ | A1 | Condone $P$ not excluded. |
| $H_0$: population medians equal or $m_1 = m_2$; $H_1$: 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 $H_0$ / not significant | M1 | Compare their '16' with their '8' and conclusion. Their '16' must be less than 23. Ignore their hypotheses. Condone 'reject $H_1$'. |
| 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. |
---
## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| The **population differences** are **symmetrical** (about the median difference) | B1 | Words in bold, or their equivalent, are required. |
---3 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.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | }
\hline
Student & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ \\
\hline
Paper 1 & 46 & 73 & 55 & 64 & 86 & 42 & 66 & 68 & 60 \\
\hline
Paper 2 & 41 & 66 & 61 & 63 & 90 & 40 & 58 & 42 & 70 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Carry out a Wilcoxon matched-pairs signed-rank test, at the $5 \%$ significance level, to test whether the data supports the teacher's belief.
\item State an assumption that you have made in carrying out the test in part (a).
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q3 [8]}}