| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2011 |
| Session | June |
| Marks | 11 |
| 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 each). Part (i) requires standard textbook knowledge about non-parametric tests. Part (ii) involves ranking 12 values, calculating rank sums, and comparing to critical values from tables—all routine procedures. Part (iii) tests understanding that adding a tied rank at the extreme doesn't change the test statistic. Slightly easier than average due to small sample size, clear data, and standard bookwork components. |
| Spec | 5.07d Paired vs two-sample: selection |
| Brand \(A :\) | 5.6 | 8.7 | 9.2 | 10.7 | 11.2 | 12.6 |
| Brand \(B :\) | 10.1 | 11.6 | 12.0 | 12.2 | 12.9 | 13.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Does not require a known probability distribution | B1 1 | Or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: m_A = m_B\), \(H_1: m_A \neq m_B\) | B1 | Medians |
| Ranks: A 1 2 3 5 6 10 | M1 | |
| B 4 7 8 9 11 12 | M1 | |
| \(R_A = 27\), \(78 - 27 = 51\), so \(W = 27\) | A1 | |
| OR: \(R_B = 51\), \(78 - 51 = 27\) | ||
| 5% CV = 26 | B1 | Use \(N(39,39)\) with cc B1 |
| \(27 > CV\) so do not reject \(H_0\) | M1 | \(P(W \leq 27.5)\), \(Z = -1.84\) or equivalent M1 |
| there is insufficient evidence at the 5% SL to indicate a difference in breaking strengths | A1 7 | Not in CR etc A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(B\) would have an extra rank 13 | M1 | |
| \(W\) still 27 but CV now 27 | B1 | \(P(W \leq 27.5) = -2.07\) M1A1 |
| \(H_0\) is now rejected | A1 3 | In CR \(H_0\) rejected A1 |
| [11] |
## Question 5:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Does not require a known probability distribution | B1 **1** | Or equivalent |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: m_A = m_B$, $H_1: m_A \neq m_B$ | B1 | Medians |
| Ranks: A 1 2 3 5 6 10 | M1 | |
| B 4 7 8 9 11 12 | M1 | |
| $R_A = 27$, $78 - 27 = 51$, so $W = 27$ | A1 | |
| OR: $R_B = 51$, $78 - 51 = 27$ | | |
| 5% CV = 26 | B1 | Use $N(39,39)$ with cc B1 |
| $27 > CV$ so do not reject $H_0$ | M1 | $P(W \leq 27.5)$, $Z = -1.84$ or equivalent M1 |
| there is insufficient evidence at the 5% SL to indicate a difference in breaking strengths | A1 **7** | Not in CR etc A1 |
### Part (iii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $B$ would have an extra rank 13 | M1 | |
| $W$ still 27 but CV now 27 | B1 | $P(W \leq 27.5) = -2.07$ M1A1 |
| $H_0$ is now rejected | A1 **3** | In CR $H_0$ rejected A1 |
| | **[11]** | |
---5 A test was carried out to compare the breaking strengths of two brands of elastic band, $A$ and $B$, of the same size. Random samples of 6 were selected from each brand and the breaking strengths were measured. The results, in suitable units and arranged in ascending order for each brand, are as follows.
\begin{center}
\begin{tabular}{ l r r r r r r }
Brand $A :$ & 5.6 & 8.7 & 9.2 & 10.7 & 11.2 & 12.6 \\
Brand $B :$ & 10.1 & 11.6 & 12.0 & 12.2 & 12.9 & 13.5 \\
\end{tabular}
\end{center}
(i) Give one advantage that a non-parametric test might have over a parametric test in this context.\\
(ii) Carry out a suitable Wilcoxon test at the $5 \%$ significance level of whether there is a difference between the average breaking strengths of the two brands.\\
(iii) An extra elastic band of brand $B$ was tested and found to have a breaking strength exceeding all of the other 12 bands. Determine whether this information alters the conclusion of your test.
\hfill \mbox{\textit{OCR S4 2011 Q5 [11]}}