| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2023 |
| Session | November |
| Marks | 10 |
| 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. Students must subtract the median (22.0) from each value, rank the absolute differences, sum positive ranks, and compare to critical values. While it requires careful arithmetic and knowledge of the test procedure, it's a standard textbook exercise with no conceptual challenges beyond executing the algorithm correctly. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07c Single-sample tests |
| 21.2 | 23.5 | 22.9 | 18.6 | 19.4 |
| 22.1 | 26.5 | 20.2 | 25.7 | 20.6 |
| 22.3 | 17.4 | 22.2 | 27.0 | 23.9 |
| 28.2 | 22.6 | 27.2 | 23.0 | 23.7 |
| 19.8 | 22.7 | 23.3 | 21.5 | 24.3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0\): population medians are equal; \(H_1\): population median after \(>\) population median before | B1 | Do not accept 'differences between population medians \(< 0\)' unless difference defined. Allow \(H_0: m=22, H_1: m > 22.0\) |
| Signed differences table with values: \(-0.8, -7, 1.5, 12, 0.9, 8, -3.4, -19, -2.6, -18\); \(0.1, 1, 4.5, 21, -1.8, -14, 3.7, 20, -1.4, -11\); \(0.3, 3, -4.6, -22, 0.2, 2, 5, 23, 1.9, 15\); \(6.2, 25, 0.6, 5, 5.2, 24, 1, 9, 1.7, 13\); \(-2.2, -16, 0.7, 6, 1.3, 10, -0.5, -4, 2.3, 17\) | M1 | Signed differences, allow at most 4 errors |
| Attempt at ranks (ignore signs) | M1 | Attempt at ranks (ignore signs). |
| \((W_+ = 214)\ \ W_- = 111\) | A1 | Cao identified, or used, as test statistic. |
| Normal: mean \(= \dfrac{1}{4}n(n+1) = \dfrac{1}{4} \times 25 \times 26 [= 162.5]\) | B1 | |
| Variance \(= \dfrac{1}{24}n(n+1)(2n+1) = \dfrac{1}{24} \times 25 \times 26 \times 51 [= 1381.25]\) | B1 | |
| \(z\)-value: \(\dfrac{111.5 - 162.5}{\sqrt{1381.25}}\) | M1 | Allow missing or incorrect continuity correction. *Their* 111 must come from ranks. |
| \(-1.37\) | A1 | CAO |
| Tabular value is \(-1.645\): \('-1.37' > -1.645\), or \(0.915 < 0.95\), oe, accept \(H_0\) | M1 | Consistent signs. Allow 'not significant'. |
| Insufficient evidence to support the teacher's claim. Insufficient evidence to suggest that the distances thrown are further when it is hot | A1 | All correct. Correct conclusion in context, following correct work, level of uncertainty in language. A0 if hypotheses wrong way round or missing |
| 10 |
# Question 6:
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0$: **population** medians are equal; $H_1$: **population** median after $>$ population median before | B1 | Do not accept 'differences between population medians $< 0$' unless difference defined. Allow $H_0: m=22, H_1: m > 22.0$ |
| Signed differences table with values: $-0.8, -7, 1.5, 12, 0.9, 8, -3.4, -19, -2.6, -18$; $0.1, 1, 4.5, 21, -1.8, -14, 3.7, 20, -1.4, -11$; $0.3, 3, -4.6, -22, 0.2, 2, 5, 23, 1.9, 15$; $6.2, 25, 0.6, 5, 5.2, 24, 1, 9, 1.7, 13$; $-2.2, -16, 0.7, 6, 1.3, 10, -0.5, -4, 2.3, 17$ | M1 | Signed differences, allow at most 4 errors |
| Attempt at ranks (ignore signs) | M1 | Attempt at ranks (ignore signs). |
| $(W_+ = 214)\ \ W_- = 111$ | A1 | Cao identified, or used, as test statistic. |
| Normal: mean $= \dfrac{1}{4}n(n+1) = \dfrac{1}{4} \times 25 \times 26 [= 162.5]$ | B1 | |
| Variance $= \dfrac{1}{24}n(n+1)(2n+1) = \dfrac{1}{24} \times 25 \times 26 \times 51 [= 1381.25]$ | B1 | |
| $z$-value: $\dfrac{111.5 - 162.5}{\sqrt{1381.25}}$ | M1 | Allow missing or incorrect continuity correction. *Their* 111 must come from ranks. |
| $-1.37$ | A1 | CAO |
| Tabular value is $-1.645$: $'-1.37' > -1.645$, or $0.915 < 0.95$, oe, accept $H_0$ | M1 | Consistent signs. Allow 'not significant'. |
| Insufficient evidence to support the teacher's claim. Insufficient evidence to suggest that the distances thrown are further when it is hot | A1 | All correct. Correct conclusion in context, following correct work, level of uncertainty in language. A0 if hypotheses wrong way round or missing |
| | **10** | |6 A school is conducting an experiment to see whether the distance that children can throw a ball increases in hot weather. On a cold day, all the children at the school were asked to throw a ball as far as possible. The distances thrown were measured and recorded. The median distance thrown by a random sample of 25 of the children was 22.0 m . The children were asked to throw the ball again on a hot day. The distances thrown by the same 25 children were measured and recorded and these distances, in m , are shown below.
\begin{center}
\begin{tabular}{ l l l l l }
21.2 & 23.5 & 22.9 & 18.6 & 19.4 \\
22.1 & 26.5 & 20.2 & 25.7 & 20.6 \\
22.3 & 17.4 & 22.2 & 27.0 & 23.9 \\
28.2 & 22.6 & 27.2 & 23.0 & 23.7 \\
19.8 & 22.7 & 23.3 & 21.5 & 24.3 \\
\end{tabular}
\end{center}
The teacher claims that on average the distances thrown will be further when it is hot.\\
Carry out a Wilcoxon signed-rank test, at the 5\% significance level, to test whether the data supports the teacher's claim.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q6 [10]}}