| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Sign test |
| Difficulty | Standard +0.3 This is a straightforward application of the sign test with clear data and standard hypothesis testing procedure. Students need to count signs above/below 2.70, apply binomial distribution, and compare to critical value. Part (ii) requires recall of the Wilcoxon signed-rank test and its properties. The question is slightly easier than average as it's a direct textbook application with no conceptual complications or multi-step reasoning required. |
| Spec | 5.07a Non-parametric tests: when to use5.07b Sign test: and Wilcoxon signed-rank |
| 3.00 | 2.05 | 3.15 | 2.65 | 3.50 | 3.25 | 2.85 | 3.35 | 2.65 | 2.75 |
| 2.90 | 2.20 | 2.95 | 3.05 | 3.65 | 3.45 | 2.55 | 2.15 | 2.80 | 2.60 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Stage 1: states 0,1,2 with minimax values 4, 3, 2 respectively | Values only credited when seen in table | |
| Stage 2, state 1: \(\max(2,4)=4\), \(\max(4,3)=4\), \(\max(5,2)=5\), minimax \(= 4\) | M1, A1 | For calculating the maxima as 4, 4, 5; for calculating the minimax as 4 |
| Stage 2, state 2: \(\max(2,4)=4\), \(\max(3,3)=3\), \(\max(4,2)=4\), minimax \(= 3\) | B1, M1, A1 | For completing 4, 3, 2 in brackets; for calculating maxima as 4, 3, 4 (method); for calculating minimax as 3, cao |
| Stage 3, state 0: \(\max(5,3)=5\), \(\max(5,4)=5\), \(\max(2,3)=3\), minimax \(= 3\) | B1, M1, A1 | For using minimax values from stage 2; for calculating maxima; for calculating minimax as 3, cao |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Value \(= 3\) | M1, A1 | For the value from their tabulation; for 3 (irrespective of tabulation), cao |
| Route: \((0;0)-(1;1)-(2;2)-(3;0)\) or in reverse | M1 dep, A1 | For reading route from their tabulation; for this route (irrespective of tabulation), cao |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Correct graph structure with nodes \((2;0)\), \((1;0)\), \((3;0)\), \((0;0)\), \((2;1)\), \((1;1)\), \((2;2)\), \((1;2)\) | B1 | For the graph structure correct |
| Substantially correct attempt at weights (no more than two definite errors or omissions) | M1 | |
| Weights unambiguously correct | A1 | |
| 3/16 |
# Question 4:
## Part (i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Stage 1: states 0,1,2 with minimax values 4, 3, 2 respectively | | Values only credited when seen in table |
| Stage 2, state 1: $\max(2,4)=4$, $\max(4,3)=4$, $\max(5,2)=5$, minimax $= 4$ | M1, A1 | For calculating the maxima as 4, 4, 5; for calculating the minimax as 4 |
| Stage 2, state 2: $\max(2,4)=4$, $\max(3,3)=3$, $\max(4,2)=4$, minimax $= 3$ | B1, M1, A1 | For completing 4, 3, 2 in brackets; for calculating maxima as 4, 3, 4 (method); for calculating minimax as 3, cao |
| Stage 3, state 0: $\max(5,3)=5$, $\max(5,4)=5$, $\max(2,3)=3$, minimax $= 3$ | B1, M1, A1 | For using minimax values from stage 2; for calculating maxima; for calculating minimax as 3, cao |
| | **4** | |
## Part (ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Value $= 3$ | M1, A1 | For the value from their tabulation; for 3 (irrespective of tabulation), cao |
| Route: $(0;0)-(1;1)-(2;2)-(3;0)$ or in reverse | M1 dep, A1 | For reading route from their tabulation; for this route (irrespective of tabulation), cao |
| | **4** | |
## Part (iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct graph structure with nodes $(2;0)$, $(1;0)$, $(3;0)$, $(0;0)$, $(2;1)$, $(1;1)$, $(2;2)$, $(1;2)$ | B1 | For the graph structure correct |
| Substantially correct attempt at weights (no more than two definite errors or omissions) | M1 | |
| Weights unambiguously correct | A1 | |
| | **3/16** | |
---4 The levels of impurity in a particular alloy were measured using a random sample of 20 specimens. The results, in suitable units, were as follows.
\begin{center}
\begin{tabular}{ l l l l l l l l l l }
3.00 & 2.05 & 3.15 & 2.65 & 3.50 & 3.25 & 2.85 & 3.35 & 2.65 & 2.75 \\
2.90 & 2.20 & 2.95 & 3.05 & 3.65 & 3.45 & 2.55 & 2.15 & 2.80 & 2.60 \\
\end{tabular}
\end{center}
(i) Use the sign test, at the $5 \%$ significance level, to decide if there is evidence that the population median level of impurity is greater than 2.70 .\\
(ii) State what other test might have been used, and give one advantage and one disadvantage this other test has over the sign test.
\hfill \mbox{\textit{OCR S4 2007 Q4 [10]}}