| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2018 |
| Session | September |
| Marks | 8 |
| 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 sample sizes (n=5 each). Part (i) requires ranking 10 values, calculating the test statistic, and comparing to critical values from tables—all standard procedure with no conceptual challenges. Part (ii) tests basic understanding of paired vs unpaired designs, requiring only a brief conceptual explanation. Slightly above average difficulty due to being a Further Maths Statistics topic, but the execution is mechanical and routine. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07d Paired vs two-sample: selection |
| Silence | 43 | 46 | 55 | 58 | 61 |
| Background music | 19 | 31 | 38 | 52 | 70 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: m_r = m_{\text{bm}}, H_1: m_r \neq m_{\text{bm}}\), where \(m_r\) and \(m_{\text{bm}}\) are the median population scores for the task in silence and with background music respectively. | B1 | 2.5 |
| Ranks \(4, 5, 7, 8, 9; 1, 2, 3, 6, 10\) | M1 | 1.1a |
| \(R_m = 22\) (or 33) | M1 | 1.1a |
| \(m(m + n + 1) - R_m = 33\) (or 22) | A1 | 1.1 |
| \(W = 22\) and CV = 19 | B1 | 1.1 |
| \(W > 19\) so do not reject \(H_0\) | M1ft | 2.2b |
| Insufficient evidence that background music affects scores | A1ft [7] | 3.5a |
| Answer | Marks | Guidance |
|---|---|---|
| Eliminates differences between students | B1 [1] | 2.3 |
## (i)
$H_0: m_r = m_{\text{bm}}, H_1: m_r \neq m_{\text{bm}}$, where $m_r$ and $m_{\text{bm}}$ are the median population scores for the task in silence and with background music respectively. | B1 | 2.5 |
Ranks $4, 5, 7, 8, 9; 1, 2, 3, 6, 10$ | M1 | 1.1a | Rank | Allow 7, 6, 4, 3, 2; 10, 9, 8, 5, 1
$R_m = 22$ (or 33) | M1 | 1.1a | Sum one set of ranks |
$m(m + n + 1) - R_m = 33$ (or 22) | A1 | 1.1 | Find $m(m+n+1) - R_m$ | If omitted can get 7/8
$W = 22$ and CV = 19 | B1 | 1.1 | Both needed |
$W > 19$ so do not reject $H_0$ | M1ft | 2.2b | FT on their $W$ only
Insufficient evidence that background music affects scores | A1ft [7] | 3.5a | FT on their $W$ only
## (ii)
Eliminates differences between students | B1 [1] | 2.3 |
---8 In an experiment to investigate the effect of background music in carrying out work, ten students were each given a task. Five of the students did the task in silence and the other five did the task with background music. The scores on the tasks were as follows.
\begin{center}
\begin{tabular}{ l l l l l l }
Silence & 43 & 46 & 55 & 58 & 61 \\
Background music & 19 & 31 & 38 & 52 & 70 \\
\end{tabular}
\end{center}
(i) Use a Wilcoxon rank-sum test to test at the 10\% level whether the presence of background music affects scores.\\
(ii) A statistician suggests that the experiment is redesigned so that each student takes one task in silence and another task with background music. The differences in the test scores would then be analysed using a paired-sample method. State an advantage in redesigning the experiment in this way.
\hfill \mbox{\textit{OCR Further Statistics 2018 Q8 [8]}}