| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Year | 2022 |
| Session | June |
| 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 Mann-Whitney U test with small sample sizes (n₁=5, n₂=5), requiring ranking data, calculating test statistic from tables, and comparing to critical values. The question is routine for Further Maths Statistics with clear structure and standard parts (justification, calculation, evaluation), though the topic itself is less commonly encountered than core A-level content, placing it slightly above average difficulty. |
| Spec | 5.07a Non-parametric tests: when to use5.07d Paired vs two-sample: selection |
| Student | Area | Stress Level |
| Heledd | North | 67 |
| Mair | North | 55 |
| Hywel | South | 26 |
| Gwyn | South | 70 |
| Liam | South | 36 |
| Marcin | South | 57 |
| Gosia | South | 32 |
| Kestutas | North | 64 |
| Erica | North | 60 |
| Tomos | North | 22 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Valid reason e.g. No knowledge of underlying distribution; Data are ordinal; Interval scale assumption may not be valid | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0\): Students from the north and the south of the county are similarly stressed. \(H_1\): Students from the north and the south of the county are NOT similarly stressed. | B1 | \(H_0: \eta_N = \eta_S\); \(H_1: \eta_N \neq \eta_S\) |
| Upper critical value \(= 22\); Lower critical value \(= 5 \times 5 - 22 = 3\) | B1 | For either CV |
| Use of the formula \(U = \sum\sum z_{ij}\); \(U = 4+0+4+3+4 = 15\) OR \(U = 1+5+1+2+1 = 10\) | M1, A1 | Attempt to use |
| Since \(15 < 22\) OR \(10 > 3\) and there is insufficient evidence to reject \(H_0\). There is not enough evidence to say that students from the North and from the South have different stress levels. | B1, E1 | FT their CV and \(U\); cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Valid improvement e.g. Bigger sample size; Use a control test to see if students from the North and South are generally more stressed | E1 |
# Question 3:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Valid reason e.g. No knowledge of underlying distribution; Data are ordinal; Interval scale assumption may not be valid | E1 | |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0$: Students from the north and the south of the county are similarly stressed. $H_1$: Students from the north and the south of the county are NOT similarly stressed. | B1 | $H_0: \eta_N = \eta_S$; $H_1: \eta_N \neq \eta_S$ |
| Upper critical value $= 22$; Lower critical value $= 5 \times 5 - 22 = 3$ | B1 | For either CV |
| Use of the formula $U = \sum\sum z_{ij}$; $U = 4+0+4+3+4 = 15$ OR $U = 1+5+1+2+1 = 10$ | M1, A1 | Attempt to use |
| Since $15 < 22$ OR $10 > 3$ and there is insufficient evidence to reject $H_0$. There is not enough evidence to say that students from the North and from the South have different stress levels. | B1, E1 | FT their CV and $U$; cso |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Valid improvement e.g. Bigger sample size; Use a control test to see if students from the North and South are generally more stressed | E1 | |
---3. A statistics teacher wants to investigate whether students from the north of a county and students from the south of the same county feel similarly stressed about examinations. The teacher carries out a psychometric test on 10 randomly selected students to give a score between 0 (low stress) and 100 (high stress) to measure their stress levels before a set of examinations. The results are shown in the table below.
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
Student & Area & Stress Level \\
\hline
Heledd & North & 67 \\
\hline
Mair & North & 55 \\
\hline
Hywel & South & 26 \\
\hline
Gwyn & South & 70 \\
\hline
Liam & South & 36 \\
\hline
Marcin & South & 57 \\
\hline
Gosia & South & 32 \\
\hline
Kestutas & North & 64 \\
\hline
Erica & North & 60 \\
\hline
Tomos & North & 22 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item State one reason why a Mann-Whitney test is appropriate.
\item Conduct a Mann-Whitney test at a significance level as close to $5 \%$ as possible. State your conclusion clearly.
\item How could this investigation be improved?
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 5 2022 Q3 [8]}}