| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Non-parametric tests |
| Type | Sign test for paired comparisons |
| Difficulty | Moderate -0.3 This is a straightforward application of the sign test with clear paired data. Students need to find differences, count positive signs, and compare to critical values from tables. While it requires understanding of hypothesis testing framework, it's a standard textbook exercise with no conceptual complications or novel problem-solving required—slightly easier than average due to its routine nature. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07c Single-sample tests |
| Student | A | \(B\) | C | D | \(E\) | \(F\) | \(G\) | H | I | J | \(K\) | \(L\) | \(M\) | \(N\) | \(O\) |
| Practical examination | 66 | 63 | 24 | 52 | 59 | 76 | 88 | 51 | 48 | 36 | 91 | 72 | 68 | 67 | 60 |
| Written examination | 63 | 57 | 39 | 50 | 47 | 71 | 87 | 65 | 56 | 39 | 78 | 70 | 61 | 62 | 70 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): practical results and written results are equal; \(H_1\): practical results are greater than written results | B1 | Or \(H_0: m_p - m_w = 0\), \(H_1: m_p - m_w > 0\), where \(m\) is population median |
| There are \(10+\) signs; \(B(15, 0.5)\) soi | M1 | |
| \(P(X \geqslant 10) = 1 - P(X \leqslant 9) = 1 - 0.849 = 0.151\) | A1 | \(P(X \leqslant 5) = 0.151\) (implies M1) |
| Compare with \(0.1\): \(0.151 > 0.1\), accept \(H_0\) | M1 | \(0.151\) must come from a valid binomial calculation. Correct ft conclusion for their \(0.151\) and \(0.1\). Condone Reject \(H_1\). Accept \(H_0\) can be implied by a conclusion consistent with their \(0.151\) and \(0.1\) |
| Insufficient evidence to suggest that practical results are greater than written results | A1 | Correct work only, ignoring their hypotheses, conclusion in context with level of uncertainty in language. Not 'prove'. Condone 'no sufficient' 'not enough'. Do not accept statements such as 'there is sufficient evidence to suggest….' |
## Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: practical results and written results are equal; $H_1$: practical results are greater than written results | B1 | Or $H_0: m_p - m_w = 0$, $H_1: m_p - m_w > 0$, where $m$ is population median |
| There are $10+$ signs; $B(15, 0.5)$ soi | M1 | |
| $P(X \geqslant 10) = 1 - P(X \leqslant 9) = 1 - 0.849 = 0.151$ | A1 | $P(X \leqslant 5) = 0.151$ (implies M1) |
| Compare with $0.1$: $0.151 > 0.1$, accept $H_0$ | M1 | $0.151$ must come from a valid binomial calculation. Correct ft conclusion for their $0.151$ and $0.1$. Condone Reject $H_1$. Accept $H_0$ can be implied by a conclusion consistent with their $0.151$ and $0.1$ |
| Insufficient evidence to suggest that practical results are greater than written results | A1 | Correct work only, ignoring their hypotheses, conclusion in context with level of uncertainty in language. Not 'prove'. Condone 'no sufficient' 'not enough'. Do not accept statements such as 'there is sufficient evidence to suggest….' |
---2 A large number of students are taking a Physics course. They are assessed by a practical examination and a written examination. The marks out of 100 obtained by a random sample of 15 students in each of the examinations are as follows.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
Student & A & $B$ & C & D & $E$ & $F$ & $G$ & H & I & J & $K$ & $L$ & $M$ & $N$ & $O$ \\
\hline
Practical examination & 66 & 63 & 24 & 52 & 59 & 76 & 88 & 51 & 48 & 36 & 91 & 72 & 68 & 67 & 60 \\
\hline
Written examination & 63 & 57 & 39 & 50 & 47 & 71 & 87 & 65 & 56 & 39 & 78 & 70 & 61 & 62 & 70 \\
\hline
\end{tabular}
\end{center}
Use a sign test, at the $10 \%$ significance level, to test whether, on average, the practical examination marks are higher than the written examination marks.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2024 Q2 [5]}}