| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2021 |
| 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 paired data. Part (a) requires standard textbook knowledge about when to use non-parametric tests, and part (b) involves routine calculation of differences, ranks, and comparison with critical values. The procedure is mechanical with no conceptual challenges beyond knowing the standard algorithm, making it slightly easier than average for Further Maths statistics. |
| Spec | 5.07a Non-parametric tests: when to use5.07b Sign test: and Wilcoxon signed-rank |
| Employee | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) | \(K\) |
| Uniform \(X\) | 82 | 74 | 42 | 59 | 60 | 73 | 94 | 98 | 62 | 36 | 50 |
| Uniform \(Y\) | 78 | 75 | 63 | 56 | 67 | 82 | 99 | 90 | 72 | 48 | 61 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Underlying distribution or population of differences is unknown; Underlying distributions or population of scores for both \(X\) and \(Y\) are unknown | B1 | Not known to be normal. Condone 'scores for \(X\) and \(Y\) cannot be assumed to be normally distributed' |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): difference of population medians \(= 0\) | B1 | Correct hypotheses, allow \(m\) but not \(\mu\) or mean |
| \(H_1\): difference of population median \(\neq 0\) | B1 | 'population' included |
| Diff: \(4\ -1\ -21\ 3\ -7\ -9\ -5\ 8\ -10\ -12\ -11\) | M1 | Differences, allow one error |
| Rank: \(3\ -1\ -11\ 2\ -5\ -7\ -4\ 6\ -8\ -10\ -9\) | M1 | Signed ranks, allow one sign error |
| \([Q = -55,]\ P = 11\) | A1 | \(P = 11\) |
| Critical tabular value \(= 13\); '\(11\)' \(< 13\) so reject \(H_0\) | M1 | Comparison with 13 and correct ft conclusion |
| Sufficient evidence of a preference for one of the uniforms OE | A1 | Correct conclusion, in context, following correct work, except possibly second B1. Level of uncertainty in language is used |
| Total | 7 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Underlying distribution or population of differences is unknown; Underlying distributions or population of scores for both $X$ and $Y$ are unknown | B1 | Not known to be normal. Condone 'scores for $X$ and $Y$ cannot be assumed to be normally distributed' |
| **Total** | **1** | |
---
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: difference of population medians $= 0$ | B1 | Correct hypotheses, allow $m$ but not $\mu$ or mean |
| $H_1$: difference of population median $\neq 0$ | B1 | 'population' included |
| Diff: $4\ -1\ -21\ 3\ -7\ -9\ -5\ 8\ -10\ -12\ -11$ | M1 | Differences, allow one error |
| Rank: $3\ -1\ -11\ 2\ -5\ -7\ -4\ 6\ -8\ -10\ -9$ | M1 | Signed ranks, allow one sign error |
| $[Q = -55,]\ P = 11$ | A1 | $P = 11$ |
| Critical tabular value $= 13$; '$11$' $< 13$ so reject $H_0$ | M1 | Comparison with 13 and correct ft conclusion |
| Sufficient evidence of a preference for one of the uniforms OE | A1 | Correct conclusion, in context, following correct work, except possibly second B1. Level of uncertainty in language is used |
| **Total** | **7** | |
---5 Georgio has designed two new uniforms $X$ and $Y$ for the employees of an airline company. A random sample of 11 employees are each asked to assess each of the two uniforms for practicality and appearance, and to give a total score out of 100. The scores are given in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Employee & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ & $J$ & $K$ \\
\hline
Uniform $X$ & 82 & 74 & 42 & 59 & 60 & 73 & 94 & 98 & 62 & 36 & 50 \\
\hline
Uniform $Y$ & 78 & 75 & 63 & 56 & 67 & 82 & 99 & 90 & 72 & 48 & 61 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Give a reason why a Wilcoxon signed-rank test may be more appropriate than a $t$-test for investigating whether there is any evidence of a preference for one of the uniforms.
\item Carry out a Wilcoxon matched-pairs signed-rank test at the $10 \%$ significance level.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2021 Q5 [8]}}