| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2017 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon rank-sum test (Mann-Whitney U test) |
| Difficulty | Standard +0.3 This is a straightforward application of the Wilcoxon rank-sum test with small samples requiring table lookup and ranking (part i), followed by a routine calculation using normal approximation for larger samples (part ii). The procedure is mechanical with no conceptual challenges beyond standard S4 material, making it slightly easier than average. |
| Spec | 5.07a Non-parametric tests: when to use5.07d Paired vs two-sample: selection |
| Girls | 1.48 | 1.31 | 1.63 | 1.38 | 1.56 | 1.57 |
| Boys | 1.44 | 1.35 | 1.32 | 1.28 | 1.27 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| G: 8 3 11 6 9 10; B: 7 5 4 2 1 | B1 | Allow 1 set |
| \(R_m = 19\), \(\{5(6+5+1)-19=41\}\), \(W=19\) | B1,B1 | Follow through incorrect ranks for these marks |
| \(H_0\): two samples come from identical populations | B1 | If in words, must say population. Or \(m_G = m_B\) and \(m_G > m_B\) |
| \(H_1\): samples do not come from identical populations | M1 | |
| "\(19\)" \(> 17\), do not reject \(H_0\) | ||
| Insufficient evidence, at 1% level, that p.s. girls are taller than p.s. boys. | A1 | cwo. Contextualised, not over-assertive |
| [6] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(N(351, 1170)\) | B1,B1 | |
| \(\dfrac{W + 0.5 - 351}{\sqrt{1170}} < -2.326\) | M1 | Allow M1 if wrong or no cc. Must be \(-\)ve z |
| B1 | 2.326 seen | |
| \(W < 270.9\) | A1 | Allow 271 for this mark, but not from 271.4, allow \(\leq\) |
| \(W = 270\) | A1 | |
| [6] |
# Question 4:
## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| G: 8 3 11 6 9 10; B: 7 5 4 2 1 | B1 | Allow 1 set |
| $R_m = 19$, $\{5(6+5+1)-19=41\}$, $W=19$ | B1,B1 | Follow through incorrect ranks for these marks |
| $H_0$: two samples come from identical populations | B1 | If in words, must say population. Or $m_G = m_B$ and $m_G > m_B$ |
| $H_1$: samples do not come from identical populations | M1 | |
| "$19$" $> 17$, do not reject $H_0$ | | |
| Insufficient evidence, at 1% level, that p.s. girls are taller than p.s. boys. | A1 | cwo. Contextualised, not over-assertive |
| | **[6]** | |
## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $N(351, 1170)$ | B1,B1 | |
| $\dfrac{W + 0.5 - 351}{\sqrt{1170}} < -2.326$ | M1 | Allow M1 if wrong or no cc. Must be $-$ve z |
| | B1 | 2.326 seen |
| $W < 270.9$ | A1 | Allow 271 for this mark, but not from 271.4, allow $\leq$ |
| $W = 270$ | A1 | |
| | **[6]** | |
---4 The heights of eleven randomly selected primary school children are measured. The results, in metres, are
\begin{center}
\begin{tabular}{ l l l l l l l }
Girls & 1.48 & 1.31 & 1.63 & 1.38 & 1.56 & 1.57 \\
Boys & 1.44 & 1.35 & 1.32 & 1.28 & 1.27 & \\
\end{tabular}
\end{center}
(i) Use a Wilcoxon rank-sum test, at the $1 \%$ significance level, to test whether primary school girls are taller than primary school boys.\\
(ii) It is decided to repeat the test, using larger random samples. The heights of twenty girls and eighteen boys are measured. Find the greatest value of the test statistic $W$ which will result in the conclusion that there is evidence, at the $1 \%$ level of significance, that primary school girls are taller than primary school boys.
\hfill \mbox{\textit{OCR S4 2017 Q4 [12]}}