| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2009 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon matched-pairs signed-rank test |
| Difficulty | Standard +0.8 This S4 question requires students to perform two different non-parametric tests (Wilcoxon signed-rank and sign test) on the same dataset, compare their conclusions, and justify which is more appropriate. The computational work is substantial, requiring calculation of differences, ranks, and test statistics for both tests, then consulting tables and interpreting results. The conceptual demand of explaining why conclusions differ and which test is preferred elevates this beyond routine application, though the individual techniques are standard for S4. |
| Spec | 5.07a Non-parametric tests: when to use5.07b Sign test: and Wilcoxon signed-rank5.07d Paired vs two-sample: selection |
| Employee | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Time on Allegro | 162 | 111 | 194 | 159 | 202 | 210 | 183 | 168 | 165 | 150 | 185 | 160 |
| Time on Vivace | 182 | 130 | 193 | 181 | 192 | 205 | 186 | 184 | 192 | 180 | 178 | 189 |
| Answer | Marks | Guidance |
|---|---|---|
| Either: State or imply \(\int\pi(2x - 3)^4 dx\) | B1 | or unsimplified equiv |
| Obtain integral of form \(k(2x-3)^5\) | M1 | any constant \(k\) involving \(\pi\) or not |
| Obtain \(\frac{\pi}{10}(2x - 3)^5\) or \(\frac{\pi}{10}\pi(2x-3)^5\) | A1 | |
| Attempt evaluation using \(0\) and \(\frac{1}{2}\) | M1 | subtraction correct way round |
| Obtain \(\frac{243}{10}\pi\) | A1 5 | or exact equiv |
| Or: State or imply \(\int\pi(2x - 3)^4 dx\) | B1 | or unsimplified equiv |
| Expand and obtain integral of order 5 | M1 | with at least three terms correct |
| Ob'n \(\frac{16}{5}x^5 - 24x^4 + 72x^2 - 108x^2 + 81x\) | A1 | with or without \(\pi\) |
| Attempt evaluation using (0 and) \(\frac{1}{2}\) | M1 | |
| Obtain \(\frac{243}{10}\pi\) | A1 (5) | or exact equiv |
Either: State or imply $\int\pi(2x - 3)^4 dx$ | B1 | or unsimplified equiv |
Obtain integral of form $k(2x-3)^5$ | M1 | any constant $k$ involving $\pi$ or not |
Obtain $\frac{\pi}{10}(2x - 3)^5$ or $\frac{\pi}{10}\pi(2x-3)^5$ | A1 |
Attempt evaluation using $0$ and $\frac{1}{2}$ | M1 | subtraction correct way round |
Obtain $\frac{243}{10}\pi$ | A1 5 | or exact equiv |
Or: State or imply $\int\pi(2x - 3)^4 dx$ | B1 | or unsimplified equiv |
Expand and obtain integral of order 5 | M1 | with at least three terms correct |
Ob'n $\frac{16}{5}x^5 - 24x^4 + 72x^2 - 108x^2 + 81x$ | A1 | with or without $\pi$ |
Attempt evaluation using (0 and) $\frac{1}{2}$ | M1 | |
Obtain $\frac{243}{10}\pi$ | A1 (5) | or exact equiv |2 A company wishes to buy a new lathe for making chair legs. Two models of lathe, 'Allegro' and 'Vivace', were trialled. The company asked 12 randomly selected employees to make a particular type of chair leg on each machine. The times, in seconds, for each employee are shown in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Employee & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
Time on Allegro & 162 & 111 & 194 & 159 & 202 & 210 & 183 & 168 & 165 & 150 & 185 & 160 \\
\hline
Time on Vivace & 182 & 130 & 193 & 181 & 192 & 205 & 186 & 184 & 192 & 180 & 178 & 189 \\
\hline
\end{tabular}
\end{center}
The company wishes to test whether there is any difference in average times for the two machines.\\
(i) State the circumstances under which a non-parametric test should be used.\\
(ii) Use two different non-parametric tests and show that they lead to different conclusions at the 5\% significance level.\\
(iii) State, with a reason, which conclusion is to be preferred.
\hfill \mbox{\textit{OCR S4 2009 Q2 [11]}}