| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2024 |
| Session | November |
| Marks | 9 |
| 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 paired data. Students must calculate differences, rank them, sum ranks for one tail, and compare to critical values from provided tables. Part (b) adds a minor twist requiring recalculation with one changed value. The procedure is mechanical and follows standard textbook methods, though it's slightly above average difficulty due to being a Further Maths topic and requiring careful arithmetic with 12 pairs. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank |
| Student | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) | \(L\) |
| Teacher 1 | 36 | 38 | 40 | 36 | 22 | 34 | 45 | 44 | 48 | 35 | 28 | 30 |
| Teacher 2 | 38 | 42 | 32 | 41 | 32 | 41 | 42 | 50 | 36 | 44 | 42 | 41 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2 \ \ 4 \ \ {-8} \ \ 5 \ \ 10 \ \ 7 \ \ {-3} \ \ 6 \ \ {-12} \ \ 9 \ \ 14 \ \ 11\) | M1 | Signed differences, allow one error. |
| \(1 \ \ 3 \ \ {-7} \ \ 4 \ \ 9 \ \ 6 \ \ {-2} \ \ 5 \ \ {-11} \ \ 8 \ \ 12 \ \ 10\) | M1 | Attempt at ranking. |
| (Sum of \(+\) ranks \(= 58\)) Test statistic \(= 20\) | A1 | |
| \(H_0\): population median for teacher \(2\) = population median for teacher \(1\); \(H_1\): population median for teacher \(2 >\) population median for teacher \(1\) | B1 | Allow use of \(m\) for population median. |
| Critical value, from tables, is 17 | B1 | |
| '\(20\)' \(> 17\), accept \(H_0\) | M1 | |
| Insufficient evidence to support the claim; insufficient evidence that the scores of Teacher 2 are higher than those of Teacher 1 | A1 | Correct work only except possibly hypotheses, in context, level of uncertainty in language. |
| Total: 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| New test statistic is '13' | B1 FT | Their 20 minus 7 |
| \(13 < 17\), so conclusion is now 'reject \(H_0\)', (sufficient evidence to support claim) | B1 | Must be 13 and 17 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2 \ \ 4 \ \ {-8} \ \ 5 \ \ 10 \ \ 7 \ \ {-3} \ \ 6 \ \ {-12} \ \ 9 \ \ 14 \ \ 11$ | **M1** | Signed differences, allow one error. |
| $1 \ \ 3 \ \ {-7} \ \ 4 \ \ 9 \ \ 6 \ \ {-2} \ \ 5 \ \ {-11} \ \ 8 \ \ 12 \ \ 10$ | **M1** | Attempt at ranking. |
| (Sum of $+$ ranks $= 58$) Test statistic $= 20$ | **A1** | |
| $H_0$: population median for teacher $2$ = population median for teacher $1$; $H_1$: population median for teacher $2 >$ population median for teacher $1$ | **B1** | Allow use of $m$ for population median. |
| Critical value, from tables, is 17 | **B1** | |
| '$20$' $> 17$, accept $H_0$ | **M1** | |
| Insufficient evidence to support the claim; insufficient evidence that the scores of Teacher 2 are higher than those of Teacher 1 | **A1** | Correct work only except possibly hypotheses, in context, level of uncertainty in language. |
| **Total: 7** | | |
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| New test statistic is '13' | B1 FT | Their 20 minus 7 |
| $13 < 17$, so conclusion is now 'reject $H_0$', (sufficient evidence to support claim) | B1 | Must be 13 and 17 |
---2 A school with a large number of students is updating its logo. Each student has designed a new logo and two teachers have each awarded a mark out of 50 for each logo. The marks awarded to a random sample of 12 students are shown in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Student & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ & $J$ & $K$ & $L$ \\
\hline
Teacher 1 & 36 & 38 & 40 & 36 & 22 & 34 & 45 & 44 & 48 & 35 & 28 & 30 \\
\hline
Teacher 2 & 38 & 42 & 32 & 41 & 32 & 41 & 42 & 50 & 36 & 44 & 42 & 41 \\
\hline
\end{tabular}
\end{center}
One of the students claims that Teacher 2 is awarding higher marks than Teacher 1.
\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 claim.\\
\includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-04_2720_38_109_2010}\\
\includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-05_2717_29_105_22}
It was later discovered that Teacher 1 had entered her mark for student $C$ incorrectly. Her intended mark was 24 not 40 . This was corrected.
\item Determine whether this correction affects the conclusion of the test carried out in part (a).
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2024 Q2 [9]}}