| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Non-parametric tests |
| Type | Wilcoxon signed-rank test |
| Difficulty | Standard +0.3 This is a straightforward application of two standard non-parametric tests with clear instructions. Students must calculate differences, rank them, and compare to critical values from tables. While it requires careful arithmetic and knowledge of both tests, it involves no conceptual difficulty or novel insight—just methodical execution of learned procedures with small sample size (n=11) making calculations manageable. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07c Single-sample tests |
| 5.56 | 5.45 | 5.47 | 5.58 | 5.54 | 5.52 | 5.60 | 5.35 | 5.59 | 5.51 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Signs: \(+-{}-++{+}-+++\) and use Binomial | M1 | 11 signs (values not needed) allow one error |
| Consider \(B(11, 0.5)\) with \(P(X \geq 8)\) or \(P(X \leq 3)\) | M1 | Use of correct binomial |
| Compare \(0.227\) with \(0.1\) or \(0.113\) with \(0.05\) | A1 | |
| Differences: 0.06, \(-0.05\), \(-0.03\), 0.08, 0.04, 0.02, 0.10, \(-0.15\), 0.09, 0.01, 0.12 | M1 | Allow one error |
| Signed Ranks: 6, \(-5\), \(-3\), 7, 4, 2, 9, \(-11\), 8, 1, 10 | M1 | Allow one swap |
| \((P=47)\ Q=19\) | A1 | |
| Critical value from table \(= 13\) | B1 | |
| \(19 > 13\); Accept \(H_0\) | M1 | FT *their* (19) compared with 13 and correct FT conclusion |
| Insufficient evidence to support median not being 5.50 | A1 | BOTH conclusions correct from correct working, in context. Level of uncertainty in language |
## Question 5:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Signs: $+-{}-++{+}-+++$ and use Binomial | M1 | 11 signs (values not needed) allow one error |
| Consider $B(11, 0.5)$ with $P(X \geq 8)$ or $P(X \leq 3)$ | M1 | Use of correct binomial |
| Compare $0.227$ with $0.1$ or $0.113$ with $0.05$ | A1 | |
| Differences: 0.06, $-0.05$, $-0.03$, 0.08, 0.04, 0.02, 0.10, $-0.15$, 0.09, 0.01, 0.12 | M1 | Allow one error |
| Signed Ranks: 6, $-5$, $-3$, 7, 4, 2, 9, $-11$, 8, 1, 10 | M1 | Allow one swap |
| $(P=47)\ Q=19$ | A1 | |
| Critical value from table $= 13$ | B1 | |
| $19 > 13$; Accept $H_0$ | M1 | FT *their* (19) compared with 13 and correct FT conclusion |
| Insufficient evidence to support median not being 5.50 | A1 | BOTH conclusions correct from correct working, in context. Level of uncertainty in language |5 A manager claims that the lengths of the rubber tubes that his company produces have a median of 5.50 cm . The lengths, in cm , of a random sample of 11 tubes produced by this company are as follows.
\begin{center}
\begin{tabular}{ l l l l l l l l l l }
5.56 & 5.45 & 5.47 & 5.58 & 5.54 & 5.52 & 5.60 & 5.35 & 5.59 & 5.51 \\
\end{tabular}
\end{center}
It is required to test at the $10 \%$ significance level the null hypothesis that the population median length is 5.50 cm against the alternative hypothesis that the population median length is not equal to 5.50 cm . Show that both a sign test and a Wilcoxon signed-rank test give the same conclusion and state this conclusion.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2022 Q5 [9]}}