| 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, and compare to critical values from provided tables. Part (b) adds minimal complexity by requiring recalculation with one changed value. The procedure is entirely algorithmic with no conceptual insight required beyond following the standard test protocol taught in Further Statistics. |
| 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 |
|---|---|---|
| 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):
| 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]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-04_2715_38_109_2010}\\
\includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-05_2716_29_107_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]}}