| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon signed-rank test (single sample) |
| Difficulty | Moderate -0.5 This is a straightforward application of the Wilcoxon signed-rank test with clear data and standard procedure. Students must calculate differences from the median, rank absolute values, sum ranks, and compare to critical values. While it requires careful execution of multiple steps, it's a routine textbook exercise with no conceptual challenges or novel problem-solving required—easier than average A-level questions. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07c Single-sample tests |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Signed differences: \(0.4\ -1.1\ 0.8\ -0.5\ 1.2\ 0.9\ 1.9\ -0.3\ -2.2\ 1.6\ 0.6\ 1.0\) | M1 | Allow one error |
| Ranks: \(2\ -8\ 5\ -3\ 9\ 6\ 11\ -1\ -12\ 10\ 4\ 7\) | M1 | Attempt at ranking |
| \(P_+ = 54\), \(P_- = 24\) | A1 | All ranks must be correct |
| \(H_0\): population median \(= 22.0\); \(H_1\): population median \(\neq 22.0\) | B1 | Accept \(m\) used as 'population median' |
| Critical value from table \(= 17\) | B1 | |
| \(24 > 17\), Accept \(H_0\) | M1 | \(24\) must come from a ranking of signed differences; \(17\) must be critical value from tables \((7, 9, 10, 13, 17\) or \(21)\). Correct ft conclusion for their \(24\) and their \(17\). Condone Reject \(H_1\). Accept \(H_0\) can be implied by a conclusion consistent with their \(24\) and their \(17\) |
| Insufficient evidence to support (population) median not being \(22.0\); Insufficient evidence against the manager's claim | A1 | Correct work only, ignoring their hypotheses, conclusion in context with level of uncertainty in language. Not 'prove'. Condone 'no sufficient' 'not enough'. Do not accept statements such as 'there is sufficient evidence to suggest….' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Distribution is symmetric about the population median; Underlying/population distribution is symmetric about the median | B1 | Need to see population or underlying distribution mentioned. Underlying distribution is symmetric B0. Population distribution is symmetric B0. 'data' implies B0 but condone 'population data'. 'mean' implies B0 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Signed differences: $0.4\ -1.1\ 0.8\ -0.5\ 1.2\ 0.9\ 1.9\ -0.3\ -2.2\ 1.6\ 0.6\ 1.0$ | M1 | Allow one error |
| Ranks: $2\ -8\ 5\ -3\ 9\ 6\ 11\ -1\ -12\ 10\ 4\ 7$ | M1 | Attempt at ranking |
| $P_+ = 54$, $P_- = 24$ | A1 | All ranks must be correct |
| $H_0$: population median $= 22.0$; $H_1$: population median $\neq 22.0$ | B1 | Accept $m$ used as 'population median' |
| Critical value from table $= 17$ | B1 | |
| $24 > 17$, Accept $H_0$ | M1 | $24$ must come from a ranking of signed differences; $17$ must be critical value from tables $(7, 9, 10, 13, 17$ or $21)$. Correct ft conclusion for their $24$ and their $17$. Condone Reject $H_1$. Accept $H_0$ can be implied by a conclusion consistent with their $24$ and their $17$ |
| Insufficient evidence to support (population) median not being $22.0$; Insufficient evidence against the manager's claim | A1 | Correct work only, ignoring their hypotheses, conclusion in context with level of uncertainty in language. Not 'prove'. Condone 'no sufficient' 'not enough'. Do not accept statements such as 'there is sufficient evidence to suggest….' |
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## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Distribution is symmetric about the population median; Underlying/population distribution is symmetric about the median | B1 | Need to see population or underlying distribution mentioned. Underlying distribution is symmetric B0. Population distribution is symmetric B0. 'data' implies B0 but condone 'population data'. 'mean' implies B0 |
---3 A factory produces metal discs. The manager claims that the diameters of these discs have a median of 22.0 mm . The diameters, in mm , of a random sample of 12 discs produced by this factory are as follows.
$$\begin{array} { l l l l l l l l l l l l }
22.4 & 20.9 & 22.8 & 21.5 & 23.2 & 22.9 & 23.9 & 21.7 & 19.8 & 23.6 & 22.6 & 23.0
\end{array}$$
\begin{enumerate}[label=(\alph*)]
\item Carry out a Wilcoxon signed-rank test, at the $10 \%$ significance level, to test whether there is any evidence against the manager's claim.
\item State an assumption that is necessary for this test to be valid.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2024 Q3 [8]}}