| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon matched-pairs signed-rank test |
| Difficulty | Standard +0.3 This is a straightforward application of the Wilcoxon matched-pairs signed-rank test with clear data and standard procedure. Students must calculate differences, rank them, sum ranks, and compare to critical values from tables. While it requires careful arithmetic and knowledge of the test procedure, it's a routine bookwork question with no conceptual challenges or novel problem-solving required—slightly easier than average for A-level Further Maths Statistics. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank |
| Student | A | B | C | D | E | F | G | H | I | J |
| Mark on paper 1 (non-calculator) | 66 | 79 | 58 | 87 | 67 | 55 | 75 | 62 | 50 | 84 |
| Mark on paper 2 (calculator) | 57 | 84 | 70 | 90 | 75 | 42 | 82 | 72 | 65 | 82 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: m_1 = m_2\), \(H_1: m_2 > m_1\) | B1 | Allow equivalent hypotheses using differences. If in words, needs 'population'. NOT: marks, NOT: papers, NOT: mean |
| Differences signs: \(-,+,+,+,+,-,+,+,+,-\) with ranks \(9,5,12,3,8,13,7,10,15,2\) and \(6,3,8,2,5,9,4,7,10,1\) | M1, A1 | 1st A1 is for correct differences |
| \(T^+ = 39\); \(T^- = 16\); \(T = 16\) | A1 | 2nd A1 is for correct \(T\) from correct ranks |
| \(CV = 10\) | B1 | |
| \(TS > CV\), do not reject \(H_0\) | M1 | ft TS, CV |
| Insufficient evidence that the calculator paper was easier. oe | A1 | ft TS. Contextualised, not over-assertive. NOT: difference, unless clearly 2-tail |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Differences symmetrical | B1 |
# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: m_1 = m_2$, $H_1: m_2 > m_1$ | B1 | Allow equivalent hypotheses using differences. If in words, needs 'population'. NOT: marks, NOT: papers, NOT: mean |
| Differences signs: $-,+,+,+,+,-,+,+,+,-$ with ranks $9,5,12,3,8,13,7,10,15,2$ and $6,3,8,2,5,9,4,7,10,1$ | M1, A1 | 1st A1 is for correct differences |
| $T^+ = 39$; $T^- = 16$; $T = 16$ | A1 | 2nd A1 is for correct $T$ from correct ranks |
| $CV = 10$ | B1 | |
| $TS > CV$, do not reject $H_0$ | M1 | ft TS, CV |
| Insufficient evidence that the calculator paper was easier. oe | A1 | ft TS. Contextualised, not over-assertive. NOT: difference, unless clearly 2-tail |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Differences symmetrical | B1 | |
---1 A teacher believes that the calculator paper in a GCSE Mathematics examination was easier than the non-calculator paper. The marks of a random sample of ten students are shown in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | }
\hline
Student & A & B & C & D & E & F & G & H & I & J \\
\hline
Mark on paper 1 (non-calculator) & 66 & 79 & 58 & 87 & 67 & 55 & 75 & 62 & 50 & 84 \\
\hline
Mark on paper 2 (calculator) & 57 & 84 & 70 & 90 & 75 & 42 & 82 & 72 & 65 & 82 \\
\hline
\end{tabular}
\end{center}
(i) Use a Wilcoxon signed-rank test, at the $5 \%$ significance level, to test the teacher's belief.\\
(ii) State the assumption necessary for this test to be applied.
\hfill \mbox{\textit{OCR S4 2014 Q1 [8]}}