| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon rank-sum test (Mann-Whitney U test) |
| Difficulty | Standard +0.3 This is a straightforward application of the Wilcoxon rank-sum test with small equal sample sizes (n=8 each). Part (a) requires ranking 16 values, calculating the test statistic, and comparing to critical values from tables—all standard procedure with no conceptual challenges. Part (b) tests basic understanding of when paired tests are appropriate (data must be paired, which these clearly aren't). This is easier than average A-level work because it's purely procedural with no problem-solving or insight required, though the non-parametric statistics topic itself is Further Maths level. |
| Spec | 5.07d Paired vs two-sample: selection |
| Test \(A\) | 46 | 32 | 29 | 12 | 33 | 18 | 25 | 40 |
| Test \(B\) | 36 | 28 | 49 | 37 | 48 | 35 | 41 | 31 |
| Answer | Marks | Guidance |
|---|---|---|
| 12 | 1 | 28 |
| 18 | 2 | 31 |
| 25 | 3 | 35 |
| 29 | 5 | 36 |
| 32 | 7 | 37 |
| 33 | 8 | 41 |
| 40 | 12 | 48 |
| 46 | 14 | 49 |
| M1 | Attempt at ranking. | |
| Test statistic: 52 | A1 | |
| \(H_0: m_x = m_y\) and \(H_1: m_x \neq m_y\) | B1 | Allow in words but 'population' must be included. |
| Critical value for \((8, 8)\) is 49 | *B1 | Allow 51 if clearly one-tail test in hypotheses. |
| \(52 > 49\); Accept \(H_0\) | DM1 | Compare their calculated value with 49 and correct FT conclusion. |
| Insufficient evidence of difference in medians | A1 | Correct conclusion, in context, following correct work. Level of uncertainty in language is used. |
| Answer | Marks | Guidance |
|---|---|---|
| Not a paired sample | B1 | |
| Underlying distribution/population not (known to be) normal; Underlying distribution/population unknown | B1 | B0 for 'data is not normally distributed'. B0 for 'marks are not normally distributed' |
## Question 4(a):
| 12 | 1 | 28 | 4 |
| 18 | 2 | 31 | 6 |
| 25 | 3 | 35 | 9 |
| 29 | 5 | 36 | 10 |
| 32 | 7 | 37 | 11 |
| 33 | 8 | 41 | 13 |
| 40 | 12 | 48 | 15 |
| 46 | 14 | 49 | 16 |
| M1 | Attempt at ranking.
Test statistic: 52 | A1 |
$H_0: m_x = m_y$ and $H_1: m_x \neq m_y$ | B1 | Allow in words but 'population' must be included.
Critical value for $(8, 8)$ is 49 | *B1 | Allow 51 if clearly one-tail test in hypotheses.
$52 > 49$; Accept $H_0$ | DM1 | Compare their calculated value with 49 and correct FT conclusion.
Insufficient evidence of difference in medians | A1 | Correct conclusion, in context, following correct work. Level of uncertainty in language is used.
---
## Question 4(b):
Not a paired sample | B1 |
Underlying distribution/population not (known to be) normal; Underlying distribution/population unknown | B1 | B0 for 'data is not normally distributed'. B0 for 'marks are not normally distributed'
---4 Applicants for a particular college take a written test when they attend for interview. There are two different written tests, $A$ and $B$, and each applicant takes one or the other. The interviewer wants to determine whether the medians of the distribution of marks obtained in the two tests are equal. The marks obtained by a random sample of 8 applicants who took test $A$ and a random sample of 8 applicants who took test $B$ are as follows.
\begin{center}
\begin{tabular}{ l l l l l l l l l }
Test $A$ & 46 & 32 & 29 & 12 & 33 & 18 & 25 & 40 \\
Test $B$ & 36 & 28 & 49 & 37 & 48 & 35 & 41 & 31 \\
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Carry out a Wilcoxon rank-sum test at the $5 \%$ significance level to determine whether there is a difference in the population median marks obtained in the two tests.\\
The interviewer considers using the given information to carry out a paired sample $t$-test to determine whether there is a difference in the population means for the two tests.
\item Give two reasons why it is not appropriate to use this test.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2021 Q4 [8]}}