| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2023 |
| Session | June |
| 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 data and standard hypotheses. Part (a) requires calculating differences, ranking absolute values, summing ranks (routine procedure), and comparing to critical values. Part (b) asks students to recalculate one rank and check if the conclusion changes—mechanical rather than conceptually demanding. While it's a Further Maths topic, the execution is algorithmic with no novel problem-solving required, making it easier than average A-level difficulty. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07c Single-sample tests |
| Company | A | B | C | D | \(E\) | \(F\) | G | \(H\) | I | J | \(K\) | \(L\) | M |
| Number in 2018 | 104 | 19 | 126 | 234 | 970 | 514 | 35 | 149 | 429 | 12 | 86 | 304 | 1104 |
| Number in 2022 | 106 | 24 | 127 | 228 | 1012 | 525 | 32 | 156 | 449 | 24 | 78 | 294 | 1154 |
| Answer | Marks | Guidance |
|---|---|---|
| Differences: \(2, 5, 1, -6, 42, 11, -3, 7, 20, 12, -8, -10, 50\) | M1 | Differences (allow up to 3 errors) |
| Ranks: \(2, 4, 1, -5, 12, 9, -3, 6, 11, 10, -7, -8, 13\) | A1 | Correct rank order, ignore signs |
| \([P = 68]\), \(Q = 23\) | A1 | CWO |
| \(H_0\): difference in population medians \(= 0\); \(H_1\): population median in 2022 \(>\) population median in 2018 | B1 | 'Population' required. Accept use of \(m\), not \(\mu\). Do not accept 'difference between population medians \(> 0\)' without 2022, 2018 OE specified |
| Critical value 21 | B1 | |
| \(23 > 21\) and accept \(H_0\) | M1 | Compare their 23 with their 21 and FT conclusion. Their 23 must be less than 46 |
| Insufficient evidence to support researcher's belief / insufficient evidence that the (median) number of employees in 2022 is greater than the (median) number of employees in 2018 | A1 | Correct conclusion in context, following correct work, level of uncertainty in language (not 'prove'), not 'there is no evidence'), no contradictions. A0 if hypotheses reversed |
| Answer | Marks | Guidance |
|---|---|---|
| Rank for \(G\) will now be \(+3\), giving \(Q = 20\) which is \(< 21\) | M1 | Must include numbers, ft their 21 from part (a) and ft their \(20\ (23-3)\) |
| Change in conclusion | A1 | CWO. Condone 'reject \(H_0\)' OE |
## Question 4(a):
| Differences: $2, 5, 1, -6, 42, 11, -3, 7, 20, 12, -8, -10, 50$ | M1 | Differences (allow up to 3 errors) |
|---|---|---|
| Ranks: $2, 4, 1, -5, 12, 9, -3, 6, 11, 10, -7, -8, 13$ | A1 | Correct rank order, ignore signs |
| $[P = 68]$, $Q = 23$ | A1 | CWO |
| $H_0$: difference in population medians $= 0$; $H_1$: population median in 2022 $>$ population median in 2018 | B1 | 'Population' required. Accept use of $m$, not $\mu$. Do not accept 'difference between population medians $> 0$' without 2022, 2018 OE specified |
| Critical value 21 | B1 | |
| $23 > 21$ and accept $H_0$ | M1 | Compare their 23 with their 21 and FT conclusion. Their 23 must be less than 46 |
| Insufficient evidence to support researcher's belief / insufficient evidence that the (median) number of employees in 2022 is greater than the (median) number of employees in 2018 | A1 | Correct conclusion in context, following correct work, level of uncertainty in language (not 'prove'), not 'there is no evidence'), no contradictions. A0 if hypotheses reversed |
---
## Question 4(b):
| Rank for $G$ will now be $+3$, giving $Q = 20$ which is $< 21$ | M1 | Must include numbers, ft their 21 from part (a) and ft their $20\ (23-3)$ |
|---|---|---|
| Change in conclusion | A1 | CWO. Condone 'reject $H_0$' OE |
---4 A random sample of 13 technology companies is chosen and the numbers of employees in 2018 and in 2022 are recorded.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
Company & A & B & C & D & $E$ & $F$ & G & $H$ & I & J & $K$ & $L$ & M \\
\hline
Number in 2018 & 104 & 19 & 126 & 234 & 970 & 514 & 35 & 149 & 429 & 12 & 86 & 304 & 1104 \\
\hline
Number in 2022 & 106 & 24 & 127 & 228 & 1012 & 525 & 32 & 156 & 449 & 24 & 78 & 294 & 1154 \\
\hline
\end{tabular}
\end{center}
A researcher claims that there has been an increase in the median number of employees at technology companies between 2018 and 2022.
\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 this claim.\\
The researcher notices that the figures for company $G$ have been recorded incorrectly. In fact, the number of employees in 2018 was 32 and the number of employees in 2022 was 35.
\item Explain, with numerical justification, whether or not the conclusion of the test in part (a) remains the same.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q4 [9]}}