| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Sign test |
| Difficulty | Moderate -0.8 This is a straightforward application of the sign test with clear data and standard hypothesis testing procedure. Part (i) requires counting signs and comparing to critical values from tables (routine S4 content), while part (ii) tests basic understanding of test comparison. The question is easier than average A-level as it's purely procedural with no problem-solving or novel insight required. |
| Spec | 5.07a Non-parametric tests: when to use5.07b Sign test: and Wilcoxon signed-rank5.07c Single-sample tests |
| Archer | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) |
| Bow type \(P\) | 95 | 97 | 92 | 85 | 87 | 92 | 90 | 89 | 98 | 77 |
| Bow type \(Q\) | 91 | 91 | 88 | 90 | 80 | 88 | 93 | 85 | 94 | 84 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: p = \frac{1}{2}\), \(H_1: p > \frac{1}{2}\) | B1 | For both. Allow any sensible hypotheses |
| Find signs of differences | M1 | \(+++-++-++-\) or vv or vv |
| Obtain 7+, 3- | A1 | |
| Attempt \(P(X \geq 7)\) or \(P(X \leq 3)\) | M1ft | |
| 0.1719 | A1ft | Allow 0.172 (0.0547 from 8+) |
| "0.1719" \(> 0.05\), so do not reject \(H_0\) | M1 | Ft candidate's \(p\). In context, not over-assertive. Cwo |
| Insufficient evidence that type P is better | A1 | Attempt to find CR. M1 (not ft) \(X \geq 9\) or \(X \leq 1\) A1 (not ft); "7" (or "3") not in CR, so d.n.r. \(H_0\) ft; NOT "suff evidence that there is no difference between the bows." |
| [7] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Magnitude of differences taken into account | B1 | Uses more information. More powerful |
| [1] |
# Question 1:
## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: p = \frac{1}{2}$, $H_1: p > \frac{1}{2}$ | B1 | For both. Allow any sensible hypotheses |
| Find signs of differences | M1 | $+++-++-++-$ or vv or vv |
| Obtain 7+, 3- | A1 | |
| Attempt $P(X \geq 7)$ or $P(X \leq 3)$ | M1ft | |
| 0.1719 | A1ft | Allow 0.172 (0.0547 from 8+) |
| "0.1719" $> 0.05$, so do not reject $H_0$ | M1 | Ft candidate's $p$. In context, not over-assertive. Cwo |
| Insufficient evidence that type P is better | A1 | Attempt to find CR. M1 (not ft) $X \geq 9$ or $X \leq 1$ A1 (not ft); "7" (or "3") not in CR, so d.n.r. $H_0$ ft; NOT "suff evidence that there is no difference between the bows." |
| **[7]** | | |
## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Magnitude of differences taken into account | B1 | Uses more information. More powerful |
| **[1]** | | |
---1 Ten archers shot at targets with two types of bow. Their scores out of 100 are shown in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | }
\hline
Archer & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ & $I$ & $J$ \\
\hline
Bow type $P$ & 95 & 97 & 92 & 85 & 87 & 92 & 90 & 89 & 98 & 77 \\
\hline
Bow type $Q$ & 91 & 91 & 88 & 90 & 80 & 88 & 93 & 85 & 94 & 84 \\
\hline
\end{tabular}
\end{center}
(i) Use the sign test, at the $5 \%$ level of significance, to test the hypothesis that bow type $P$ is better than bow type $Q$.\\
(ii) Why would a Wilcoxon signed rank test, if valid, be a better test than the sign test?
\hfill \mbox{\textit{OCR S4 2016 Q1 [8]}}