| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2024 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon signed-rank test (single sample) |
| Difficulty | Standard +0.3 This is a straightforward application of the Wilcoxon signed-rank test with clear instructions and provided critical value tables. Students need to calculate differences from the median, rank absolute differences, sum ranks for the appropriate tail, and compare to the table. While it requires careful arithmetic and understanding of the test procedure, it follows a standard algorithm with no conceptual surprises, making it slightly easier than average for Further Maths statistics. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank |
| 51.3 | 52.0 | 53.4 | 49.2 | 49.3 | 51.1 | 52.2 | 47.2 |
| 53.0 | 48.5 | 49.4 | 50.3 | 50.8 | 51.6 | 49.1 | 52.3 |
| 51.8 | 52.4 | 47.9 | 48.9 | 50.6 | 51.9 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Signed differences from 51.0: \(0.3, 1.0, 2.4, -1.8, -1.7, 0.1, 1.2, -3.8, 2.0, -2.5, -1.6, -0.7, -0.2, 0.6, -1.9, 1.3, 0.8, 1.4, -3.1, -2.1, -0.4, 0.9\) | M1 | Allow up to 2 errors |
| Ranks of signed differences: \(3, 9, 19, -15, -14, 1, 10, -22, 17, -20, -13, -6, -2, 5, -16, 11, 7, 12, -21, -18, -4, 8\) | M1 | Attempt at ranking of signed differences from 51.0 |
| Smaller sum is 102 | A1 | 151 |
| \(H_0\): population median is 51.0 (seconds); \(H_1\): population median is less than 51.0 (seconds) | B1 | Allow \(m\) |
| Normal approximation: mean \(= 126.5\), variance \(= 948.75\) | B1 | Both |
| \(\dfrac{102 + 0.5 - 126.5}{\sqrt{948.75}}\) | M1 | Allow no continuity correction |
| \(-0.779\) | A1 | Accept \(0.779\). Accept \(p\)-value \(= 0.218\). (A0 for \(z = -0.795\) or \(p = 0.213\) from no continuity correction) |
| \(-0.779 > -1.645\), accept \(H_0\) | M1 | Or \('0.218' > 0.05\) or \('0.782' < 0.95\) |
| Insufficient evidence to suggest that (population) median is less than 51.0 seconds / Insufficient evidence to support the claim | A1 | Correct work only, except possibly hypotheses, in context, level of uncertainty in language. Alternative approach using critical values. |
| Total | 9 |
## Question 6:
| Answer | Mark | Guidance |
|--------|------|----------|
| Signed differences from 51.0: $0.3, 1.0, 2.4, -1.8, -1.7, 0.1, 1.2, -3.8, 2.0, -2.5, -1.6, -0.7, -0.2, 0.6, -1.9, 1.3, 0.8, 1.4, -3.1, -2.1, -0.4, 0.9$ | M1 | Allow up to 2 errors |
| Ranks of signed differences: $3, 9, 19, -15, -14, 1, 10, -22, 17, -20, -13, -6, -2, 5, -16, 11, 7, 12, -21, -18, -4, 8$ | M1 | Attempt at ranking of signed differences from 51.0 |
| Smaller sum is 102 | A1 | 151 |
| $H_0$: population median is 51.0 (seconds); $H_1$: population median is less than 51.0 (seconds) | B1 | Allow $m$ |
| Normal approximation: mean $= 126.5$, variance $= 948.75$ | B1 | Both |
| $\dfrac{102 + 0.5 - 126.5}{\sqrt{948.75}}$ | M1 | Allow no continuity correction |
| $-0.779$ | A1 | Accept $0.779$. Accept $p$-value $= 0.218$. (A0 for $z = -0.795$ or $p = 0.213$ from no continuity correction) |
| $-0.779 > -1.645$, accept $H_0$ | M1 | Or $'0.218' > 0.05$ or $'0.782' < 0.95$ |
| Insufficient evidence to suggest that (population) median is less than 51.0 seconds / Insufficient evidence to support the claim | A1 | Correct work only, except possibly hypotheses, in context, level of uncertainty in language. Alternative approach using critical values. |
| **Total** | **9** | |6 A sports college keeps records of the times taken by students to run one lap of a running track. The population median time taken is 51.0 seconds. After a month of intensive training, a random sample of 22 new students run one lap of the track, giving times, in seconds, as follows.
\begin{center}
\begin{tabular}{ c c c c c c c c }
51.3 & 52.0 & 53.4 & 49.2 & 49.3 & 51.1 & 52.2 & 47.2 \\
53.0 & 48.5 & 49.4 & 50.3 & 50.8 & 51.6 & 49.1 & 52.3 \\
51.8 & 52.4 & 47.9 & 48.9 & 50.6 & 51.9 & & \\
\end{tabular}
\end{center}
It is claimed that the intensive training has led to a decrease in the median time taken to run one lap of the track.
Carry out a Wilcoxon signed-rank test, at the $5 \%$ significance level, to test whether there is sufficient evidence to support the claim.\\
\includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-13_2726_35_97_20}\\
If you use the following page to complete the answer to any question, the question number must be clearly shown.\\
\includegraphics[max width=\textwidth, alt={}, center]{b9cbf607-4f40-41bb-8374-6b2c39f945ac-14_2715_33_109_2012}
\hfill \mbox{\textit{CAIE Further Paper 4 2024 Q6 [9]}}