| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| 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 matched-pairs signed-rank test with clear paired data and a one-tailed hypothesis. Students must calculate differences, rank absolute values, sum ranks for negative differences, and compare to critical values from tables. While it requires careful arithmetic and understanding of the test procedure, it's a standard textbook exercise with no conceptual surprises—slightly easier than average since the method is prescribed and mechanical. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank |
| Patient | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Drug I | 11 | 34 | 19 | 16 | 10 | 29 | 6 | 17 | 20 | 13 | 4 | 25 |
| Drug II | 12 | 20 | 10 | 18 | 3 | 21 | 9 | 13 | 10 | 19 | 9 | 12 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: m_{\text{II}-\text{I}} = 0\), \(H_1: m_{\text{II}-\text{I}} < 0\) | B1 | Allow \(m_1 > m_2\), \(m_d > 0\) etc. or in words, but needs to be in terms of parameters or population |
| II–I: \(1, -14, -9, 2, -7, -8, 3, -4, -10, 6, 5, 13\) | M1 | |
| Rank: \(1, -12, -9, 2, -7, -8, 3, -4, -10, 6, 5, 11\) | A1 | |
| \(P = 1+2+3+6+5 = 17\) | ||
| \(Q = 61\) so \(T = 17\); 5% CR: \(T \leq 17\) | M1A1 | |
| \(T\) is inside CR so reject \(H_0\) | M1 | ft TS & CV |
| There is sufficient evidence at the 5% SL that drug II is associated with fewer sneezes | A1 | ft TS only. Contextualised, not over-assertive |
| [7] |
# Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: m_{\text{II}-\text{I}} = 0$, $H_1: m_{\text{II}-\text{I}} < 0$ | B1 | Allow $m_1 > m_2$, $m_d > 0$ etc. or in words, but needs to be in terms of parameters or population |
| II–I: $1, -14, -9, 2, -7, -8, 3, -4, -10, 6, 5, 13$ | M1 | |
| Rank: $1, -12, -9, 2, -7, -8, 3, -4, -10, 6, 5, 11$ | A1 | |
| $P = 1+2+3+6+5 = 17$ | | |
| $Q = 61$ so $T = 17$; 5% CR: $T \leq 17$ | M1A1 | |
| $T$ is inside CR so reject $H_0$ | M1 | ft TS & CV |
| There is sufficient evidence at the 5% SL that drug II is associated with fewer sneezes | A1 | ft TS only. Contextualised, not over-assertive |
| **[7]** | | |
---2 Two drugs, I and II, for alleviating hay fever are trialled in a hospital on each of 12 volunteer patients. Each received drug I on one day and drug II on a different day. After receiving a drug, the number of times each patient sneezed over a period of one hour was noted. The results are given in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Patient & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
Drug I & 11 & 34 & 19 & 16 & 10 & 29 & 6 & 17 & 20 & 13 & 4 & 25 \\
\hline
Drug II & 12 & 20 & 10 & 18 & 3 & 21 & 9 & 13 & 10 & 19 & 9 & 12 \\
\hline
\end{tabular}
\end{center}
The patients may be considered to be a random sample of all hay fever sufferers.\\
A researcher believes that patients taking drug II sneeze less than patients taking drug I.\\
Test this belief using the Wilcoxon signed rank test at the $5 \%$ significance level.
\hfill \mbox{\textit{OCR S4 2013 Q2 [7]}}