| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| 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 Wilcoxon rank-sum test with small samples (n=6, n=7) requiring ranking, calculating test statistic, and comparing to tables. Part (ii) requires recall of t-test assumptions and part (iii) needs basic understanding of test efficiency. The ranking is simple with no ties, and the question guides students through each step clearly. Slightly above average difficulty due to being Further Maths content and requiring knowledge of when different tests apply, but otherwise routine. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank |
| Lecture theatre: | 30 | 36 | 48 | 51 | 59 | 62 | |
| Filmed lecture: | 40 | 49 | 52 | 56 | 63 | 64 | 68 |
| Answer | Marks | Guidance |
|---|---|---|
| \(32 > 29\), do not reject H | B1, M1, A1, A1, B1, M1 | Allow 'Data quantitative', Can be implied, M1A0A1 possible, Correct first conclusion ft TS and CV |
| There is insufficient evidence at the 10% significance level of a difference between the median marks of the two groups. oe. | A1 | ft TS only. |
| [7] | ||
| (ii) Marks should have normal populations with equal variances. | B1 | Need 'population'. |
| [1] | ||
| (iii) 2- sample t-test would be better than the Wilcoxon test since it uses more information. | B1 | Or is more powerful. |
| [1] |
**(i)** Populations have identical/same distributions (apart from location)
$(H_0: m_1=m_2, H_1:m_1 \neq m_2 )$
Ranks: 1 2 4 6 9 10
3 5 7 8 11 12 13
$R_m = 32, m(m + n + 1) – R_m = 52$
$W = 32$
Critical value $= 29$
$32 > 29$, do not reject H | B1, M1, A1, A1, B1, M1 | Allow 'Data quantitative', Can be implied, M1A0A1 possible, Correct first conclusion ft TS and CV
There is insufficient evidence at the 10% significance level of a difference between the median marks of the two groups. oe. | A1 | ft TS only.
| [7] |
**(ii)** Marks should have normal populations with equal variances. | B1 | Need 'population'. | NOT populations of students.
| [1] |
**(iii)** 2- sample t-test would be better than the Wilcoxon test since it uses more information. | B1 | Or is more powerful.
| [1] |3 Because of the large number of students enrolled for a university geography course and the limited accommodation in the lecture theatre, the department provides a filmed lecture. Students are randomly assigned to two groups, one to attend the lecture theatre and the other the film. At the end of term the two groups are given the same examination. The geography professor wishes to test whether there is a difference in the performance of the two groups and selects the marks of two random samples of students, 6 from the group attending the lecture theatre and 7 from the group attending the films. The marks for the two samples, ordered for convenience, are shown below.
\begin{center}
\begin{tabular}{ l l l l l l l l }
Lecture theatre: & 30 & 36 & 48 & 51 & 59 & 62 & \\
Filmed lecture: & 40 & 49 & 52 & 56 & 63 & 64 & 68 \\
\end{tabular}
\end{center}
(i) Stating a necessary assumption, carry out a suitable non-parametric test, at the $10 \%$ significance level, for a difference between the median marks of the two groups.\\
(ii) State conditions under which a two-sample $t$-test could have been used.\\
(iii) Assuming that the tests in parts (i) and (ii) are both valid, state with a reason which test would be preferable.
\hfill \mbox{\textit{OCR S4 2012 Q3 [9]}}