| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon rank-sum test (Mann-Whitney U test) |
| Difficulty | Challenging +1.2 This is a standard application of the Mann-Whitney U test with clear data and straightforward procedure. While it's a Further Maths topic (non-parametric statistics), the question follows a routine template: state hypotheses, find critical region from tables, calculate U by ranking data, and compare. The calculation involves ranking 12 values and applying a formula, but requires no novel insight or complex problem-solving beyond following the standard algorithm. |
| Spec | 5.07a Non-parametric tests: when to use5.07d Paired vs two-sample: selection |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): The petrol consumptions of models A and B are the same | B1 | AO3 |
| \(H_1\): The petrol consumptions of models A and B are not the same | B1 | AO3 |
| Answer | Marks | Guidance |
|---|---|---|
| From tables upper crit value = 31 | B1 | AO1 |
| Therefore lower crit value = 36 − 31 = 5 | B1 | AO2 |
| The critical region is \((U \geq 31) \cup (U \leq 5)\) | B1 | AO2 |
| Answer | Marks | Guidance |
|---|---|---|
| Use of the formula \(U = \sum \sum z_{ij}\) | M1 | AO3 |
| \(U' = 1 + 6 + 2 + 6 + 6 + 3 = 24\) | A1 | AO1 |
| The conclusion is that there is no difference in petrol consumption of the two models because 24 is not in the critical region. | B1 | AO3 |
| B1 | AO2 | |
| [9] |
## 3(a)
$H_0$: The petrol consumptions of models A and B are the same | B1 | AO3 | B0 for saying that the mean petrol consumption is the same
$H_1$: The petrol consumptions of models A and B are not the same | B1 | AO3 | For correctly identifying the alternative hypothesis as two-sided
## 3(b)
From tables upper crit value = 31 | B1 | AO1 |
Therefore lower crit value = 36 − 31 = 5 | B1 | AO2 |
The critical region is $(U \geq 31) \cup (U \leq 5)$ | B1 | AO2 |
## 3(c)
Use of the formula $U = \sum \sum z_{ij}$ | M1 | AO3 |
$U' = 1 + 6 + 2 + 6 + 6 + 3 = 24$ | A1 | AO1 |
The conclusion is that there is no difference in petrol consumption of the two models because 24 is not in the critical region. | B1 | AO3 |
| B1 | AO2 |
| [9] | |A motoring organisation wishes to determine whether or not the petrol consumption of two different car models A and B are the same. A trial is therefore carried out in which 6 cars of each model are given 10 litres of petrol and driven at a predetermined speed around a track until the petrol is used up. The distances travelled, in miles, are shown below
Model A: $86.3 \quad 84.2 \quad 85.8 \quad 83.1 \quad 84.7 \quad 85.3$
Model B: $84.9 \quad 85.9 \quad 84.8 \quad 86.5 \quad 85.2 \quad 85.5$
It is proposed to use a test with significance level 5% based on the Mann-Whitney statistic $U$.
\begin{enumerate}[label=(\alph*)]
\item State suitable hypotheses. [2]
\item Find the critical region for the test. [3]
\item Determine the value of $U$ for the above data and state your conclusion in context. You must justify your answer. [4]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 5 Q3 [9]}}