| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2020 |
| 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. Part (a) requires standard procedure (calculate differences, rank absolute values, sum ranks) with no conceptual challenges. Part (b) tests understanding of how changing one data point affects the test statistic, requiring only basic reasoning about rank changes. The question is slightly easier than average because it's a direct textbook application with no novel problem-solving required, though the two-part structure and need to interpret the effect of data changes adds minor complexity. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank |
| Student | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) | \(L\) |
| Judge 1 | 62 | 74 | 52 | 48 | 68 | 55 | 56 | 64 | 37 | 70 | 81 | 59 |
| Judge 2 | 65 | 70 | 47 | 49 | 76 | 74 | 67 | 54 | 50 | 77 | 72 | 75 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): difference in (population) medians \(= 0\); \(H_1\): difference in (population) medians \(< 0\) (or \(> 0\)) | B1 B1 | Correct hypotheses (median or \(m\)); all notation identified e.g. \(m\) = population median if used |
| Differences: \(-3\ 4\ 5\ -1\ -8\ -19\ -11\ 10\ -13\ -7\ 9\ -16\) | M1 | At most 3 errors |
| Ranks: \(-2\ 3\ 4\ -1\ -6\ -12\ -9\ 8\ -10\ -5\ 7\ -11\) | A1 | Award for correct rank order, ignore signs |
| \(\text{Sum } T = 3 + 4 + 8 + 7 = 22\) | A1 | cwo |
| Compare with critical value 17: \(22 > 17\) | M1 | Compare *their* T with 17 |
| Accept \(H_0\); data does not support student's claim | A1 | In context, all correct, except possibly second B1; level of uncertainty in language used. No contradictions. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Rank for A becomes \(+2\), | M1 | \(T = 24\) |
| Changing sign of difference can only reduce evidence in favour of the claim. | A1 | still \(> 17\) and test result unchanged |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: difference in (population) medians $= 0$; $H_1$: difference in (population) medians $< 0$ (or $> 0$) | B1 B1 | Correct hypotheses (median or $m$); all notation identified e.g. $m$ = population median if used |
| Differences: $-3\ 4\ 5\ -1\ -8\ -19\ -11\ 10\ -13\ -7\ 9\ -16$ | M1 | At most 3 errors |
| Ranks: $-2\ 3\ 4\ -1\ -6\ -12\ -9\ 8\ -10\ -5\ 7\ -11$ | A1 | Award for correct rank order, ignore signs |
| $\text{Sum } T = 3 + 4 + 8 + 7 = 22$ | A1 | cwo |
| Compare with critical value 17: $22 > 17$ | M1 | Compare *their* T with 17 |
| Accept $H_0$; data does not support student's claim | A1 | In context, all correct, except possibly second B1; level of uncertainty in language used. No contradictions. |
---
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Rank for A becomes $+2$, | M1 | $T = 24$ |
| Changing sign of difference can only reduce evidence in favour of the claim. | A1 | still $> 17$ and test result unchanged |
---2 A large school is holding an essay competition and each student has submitted an essay. To ensure fairness, each essay is given a mark out of 100 by two different judges. The marks awarded to the essays submitted by a random sample of 12 students are shown in the following table.
\begin{center}
\begin{tabular}{ | c | 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
Judge 1 & 62 & 74 & 52 & 48 & 68 & 55 & 56 & 64 & 37 & 70 & 81 & 59 \\
\hline
Judge 2 & 65 & 70 & 47 & 49 & 76 & 74 & 67 & 54 & 50 & 77 & 72 & 75 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item One of the students claims that Judge 2 is awarding higher marks than Judge 1.
Carry out a Wilcoxon matched-pairs signed-rank test at the $5 \%$ significance level to test whether the data supports the student's claim.\\
It is discovered later that the marks awarded to student $A$ have been entered incorrectly. In fact, Judge 1 awarded 65 marks and Judge 2 awarded 62 marks.
\item By considering how this change affects the test statistic, explain why the conclusion of the test carried out in part (a) remains the same.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2020 Q2 [9]}}