| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| 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 Mann-Whitney U test with small sample sizes (n=6 each). Part (a) requires ranking 12 values, calculating U statistics, and comparing to tables—all standard procedure with no conceptual challenges. Part (b) tests basic understanding of when paired vs unpaired tests apply. Slightly easier than average A-level as it's purely procedural with clear data and standard method. |
| Spec | 5.07a Non-parametric tests: when to use5.07b Sign test: and Wilcoxon signed-rank5.07d Paired vs two-sample: selection |
| 1 | 14 | 12 | 5 | 3 | 7 | ||
| 2 | 9 | 4 | 0 | 8 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(H_0\): The median numbers of races entered by competitors who are club members and those who are not club members are the same. | B1 |
| \(H_1\): The median number of races entered by competitors who are club members is more than the median number of races entered by those who are not club members. | ||
| Use of the formula \(U = \sum\sum z_{ij}\) | M1 | oe |
| \(U = 1 + 6 + 6 + 3 + 2 + 4\) OR \(U = 5 + 0 + 0 + 3 + 4 + 2\) | A1 | |
| \(U = 22\) OR \(U = 14\) | ||
| Upper critical value \(= 29\) OR Lower CV is 7 | B1 | |
| \(22 < 29\) OR \(14 > 7\), there is insufficient evidence to reject \(H_0\). | m1 | |
| There is insufficient evidence to suggest that athletes race more frequently if they are members of a triathlon club. | A1 | cso |
| (b) | The samples are independent rather than paired. | E1 |
| Total [7] |
(a) | $H_0$: The median numbers of races entered by competitors who are club members and those who are not club members are the same. | B1 | Accept $H_0: \eta_1 = \eta_2$ and $H_1: \eta_1 > \eta_2$ |
| $H_1$: The median number of races entered by competitors who are club members is more than the median number of races entered by those who are not club members. | | |
| | | |
| Use of the formula $U = \sum\sum z_{ij}$ | M1 | oe |
| $U = 1 + 6 + 6 + 3 + 2 + 4$ OR $U = 5 + 0 + 0 + 3 + 4 + 2$ | A1 | |
| $U = 22$ OR $U = 14$ | | |
| Upper critical value $= 29$ OR Lower CV is 7 | B1 | |
| $22 < 29$ OR $14 > 7$, there is insufficient evidence to reject $H_0$. | m1 | |
| There is insufficient evidence to suggest that athletes race more frequently if they are members of a triathlon club. | A1 | cso |
(b) | The samples are independent rather than paired. | E1 | E0 for data is ordinal. Ignore spurious additional comments |
| **Total [7]** | | |
---6. A triathlon race organiser wishes to know whether competitors who are members of a triathlon club race more frequently than competitors who are not members of a triathlon club. Six competitors from a triathlon club and six competitors who are not members of a triathlon club are selected at random. The table below shows the number of triathlon races they each entered last year.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
\begin{tabular}{ l }
Club \\
members \\
\end{tabular} & 1 & 14 & 12 & 5 & 3 & 7 \\
\hline
\begin{tabular}{ l }
Not club \\
members \\
\end{tabular} & 2 & 9 & 4 & 0 & 8 & 6 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use a Mann-Whitney U test at a significance level as close to $5 \%$ as possible to carry out the race organiser's investigation.
\item Briefly explain why a Wilcoxon signed-rank test is not appropriate in this case.
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 5 2023 Q6 [7]}}