| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2022 |
| Session | November |
| 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 paired data. Students must calculate differences, rank absolute values, sum ranks for negative differences, and compare to critical values from tables. Part (b) requires recalling when non-parametric tests are preferred. While it involves multiple computational steps, it's a standard textbook procedure with no novel insight required, making it slightly easier than average for Further Maths statistics. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07c Single-sample tests |
| Student | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) | \(L\) | \(M\) | \(N\) |
| Judge 1 | 79 | 54 | 63 | 74 | 69 | 52 | 50 | 57 | 55 | 42 | 63 | 55 | 56 | 48 |
| Judge 2 | 75 | 62 | 60 | 73 | 76 | 41 | 31 | 51 | 45 | 55 | 49 | 50 | 65 | 36 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Differences: \(4\ -8\ 3\ 1\ -7\ 11\ 19\ 6\ 10\ -13\ 14\ 5\ -9\ 12\) | M1 | Attempt at signed differences, allow 2 errors |
| Signed Ranks: \(3\ -7\ 2\ 1\ -6\ 10\ 14\ 5\ 9\ -12\ 13\ 4\ -8\ 11\) | M1 | Ranks, allow 4 errors |
| \([P = 72,\ Q = 33]\ T = 33\) | A1 | 33 clearly identified |
| \(H_0\): difference in population medians \(= 0\); \(H_1\): difference in population medians \(> 0\) | B1 | Correct, allow \(m\) for this mark |
| Critical tabular value \(= 25\) | B1 | |
| \(33 > 25\); accept \(H_0\) | M1 | Compares their value of sum of ranks with their 25 and conclusion |
| Insufficient evidence to support claim | A1 | All correct except possibly first B1, in context, level of uncertainty in language. 'Prove' scores A0 |
| 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Underlying/population distribution of differences is unknown (not known to be normal) | B1 | |
| 1 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Differences: $4\ -8\ 3\ 1\ -7\ 11\ 19\ 6\ 10\ -13\ 14\ 5\ -9\ 12$ | M1 | Attempt at signed differences, allow 2 errors |
| Signed Ranks: $3\ -7\ 2\ 1\ -6\ 10\ 14\ 5\ 9\ -12\ 13\ 4\ -8\ 11$ | M1 | Ranks, allow 4 errors |
| $[P = 72,\ Q = 33]\ T = 33$ | A1 | 33 clearly identified |
| $H_0$: difference in population medians $= 0$; $H_1$: difference in population medians $> 0$ | B1 | Correct, allow $m$ for this mark |
| Critical tabular value $= 25$ | B1 | |
| $33 > 25$; accept $H_0$ | M1 | Compares their value of sum of ranks with their 25 and conclusion |
| Insufficient evidence to support claim | A1 | All correct except possibly first B1, in context, level of uncertainty in language. 'Prove' scores A0 |
| | **7** | |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Underlying/population distribution of differences is unknown (not known to be normal) | B1 | |
| | **1** | |3 A large college is holding a piano competition. Each student has played a particular piece of music and two judges have each awarded a mark out of 80 . The marks awarded to a random sample of 14 students are shown in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Student & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ & $J$ & $K$ & $L$ & $M$ & $N$ \\
\hline
Judge 1 & 79 & 54 & 63 & 74 & 69 & 52 & 50 & 57 & 55 & 42 & 63 & 55 & 56 & 48 \\
\hline
Judge 2 & 75 & 62 & 60 & 73 & 76 & 41 & 31 & 51 & 45 & 55 & 49 & 50 & 65 & 36 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item One of the students claims that on average Judge 1 is awarding higher marks than Judge 2. Carry out a Wilcoxon matched-pairs signed-rank test at the 5\% significance level to test whether the data supports the student's claim.
\item Give a reason why it is preferable to use a Wilcoxon matched-pairs signed-rank test in this situation rather than a paired sample $t$-test.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2022 Q3 [8]}}