| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| 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 signed-rank test with clear paired data. Part (a) requires standard justification (paired data, small sample, no normality assumption). Part (b) involves routine calculation of differences, ranks, and test statistic comparison with critical values from tables. While it's a Further Maths statistics question requiring knowledge of non-parametric tests, the execution is mechanical with no conceptual challenges or novel problem-solving required. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank |
| Student | A | B | C | D | E | F | G | H |
| Experienced Examiner | 108 | 109 | 92 | 95 | 145 | 148 | 134 | 120 |
| Trainee | 114 | 116 | 95 | 93 | 137 | 144 | 133 | 110 |
| Answer | Marks | Guidance |
|---|---|---|
| Valid explanation. e.g. It is a small sample. e.g. There is no reason to suppose that there is an underlying normal distribution. e.g. The data is paired. | E1 E1 | Do not allow contradicting statements |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): There is on average no difference between scores of the trainee and experienced examiner. \(H_1\): The trainee and experienced examiners give different scores on average. OR \(H_0: \mu_1 = \mu_2\) \(H_1: \mu_1 \neq \mu_2\) | B1 | Both (alternative: \(H_0\): The scores of the trainee and experienced examiner have the same distribution. \(H_1\): The scores of the trainee and experienced examiner don't have the same distribution.) |
| Student | A | B |
| Difference | 6 | 7 |
| B1 | — | |
| Ranks | ||
| Student | A | B |
| Ranks | 5 | 6 |
| M1 A1 | Accept ranks with opposite signs. M1 either attempt at ranks. FT one slip in difference for A1 | |
| \(W = \text{Sum of negative ranks OR } W' = \text{Sum of positive ranks} = 2 + 7 + 4 + 1 + 8 = 22\) | M1 A1 | — |
| \(= 5 + 6 + 3 = 14\) | — | — |
| Upper CV \(= 32\) | B1 | — |
| Lower CV \(= 4\) | — | — |
| Because \(22 < 32\) (OR \(14 > 4\)) there is insufficient evidence to reject \(H_0\). | B1 | — |
| Answer | Marks | Guidance |
|---|---|---|
| The trainee examiner is suitable to qualify. | E1 | — |
## (a)
Valid explanation. e.g. It is a small sample. e.g. There is no reason to suppose that there is an underlying normal distribution. e.g. The data is paired. | E1 E1 | Do not allow contradicting statements
## (b)(i)
$H_0$: There is on average no difference between scores of the trainee and experienced examiner. $H_1$: The trainee and experienced examiners give different scores on average. OR $H_0: \mu_1 = \mu_2$ $H_1: \mu_1 \neq \mu_2$ | B1 | Both (alternative: $H_0$: The scores of the trainee and experienced examiner have the same distribution. $H_1$: The scores of the trainee and experienced examiner don't have the same distribution.)
| Student | A | B | C | D | E | F | G | H |
|---------|---|---|---|---|---|---|---|---|
| Difference | 6 | 7 | 3 | -2 | -8 | -4 | -1 | -10 |
| B1 | —
| Ranks | | | | | | | | |
|-------|---|---|---|---|---|---|---|---|
| Student | A | B | C | D | E | F | G | H |
| Ranks | 5 | 6 | 3 | 2 | 7 | 4 | 1 | 8 |
| M1 A1 | Accept ranks with opposite signs. M1 either attempt at ranks. FT one slip in difference for A1
$W = \text{Sum of negative ranks OR } W' = \text{Sum of positive ranks} = 2 + 7 + 4 + 1 + 8 = 22$ | M1 A1 | —
$= 5 + 6 + 3 = 14$ | — | —
Upper CV $= 32$ | B1 | —
Lower CV $= 4$ | — | —
Because $22 < 32$ (OR $14 > 4$) there is insufficient evidence to reject $H_0$. | B1 | —
## (b)(ii)
The trainee examiner is suitable to qualify. | E1 | —
**Total: [11]**
---To qualify as a music examiner, a trainee must listen to a series of performances by 8 randomly chosen students. An experienced examiner and the trainee must both award scores for each of the 8 performances. In order for the trainee to qualify, there must not be a significant difference between the average scores given by the experienced examiner and the trainee.
\begin{enumerate}[label=(\alph*)]
\item Explain why the Wilcoxon signed rank test is appropriate. [2]
\end{enumerate}
The scores awarded are shown below.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Student & A & B & C & D & E & F & G & H \\
\hline
Experienced Examiner & 108 & 109 & 92 & 95 & 145 & 148 & 134 & 120 \\
\hline
Trainee & 114 & 116 & 95 & 93 & 137 & 144 & 133 & 110 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item
\begin{enumerate}[label=(\roman*)]
\item Carry out an appropriate Wilcoxon signed rank test on this dataset, using a 5\% significance level.
\item What conclusion should be reached about the suitability of the trainee to qualify? [9]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 5 2019 Q5 [11]}}