| Exam Board | OCR |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Critical region or test statistic properties |
| Difficulty | Challenging +1.8 This question requires understanding of exact probability distributions for the Wilcoxon rank-sum test, including enumeration of outcomes and calculation of tail probabilities. Part (i) demands combinatorial reasoning to verify a specific probability, while part (ii) requires computing the test statistic and finding its exact significance level from the null distribution—going beyond standard table lookup to demonstrate deep understanding of the test's theoretical foundation. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07d Paired vs two-sample: selection |
| Rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| School | \(A\) | \(A\) | \(A\) | \(B\) | \(A\) | \(A\) | \(B\) | \(B\) | \(B\) | \(B\) | \(B\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Number of different rankings \({}^{11}C_5\) | M1 | Number of selections of 5 from 11 |
| \(= 462\) | A1 | |
| For \(R \leq 17\): \(1+2+3+4+5=15\); \(1+2+3+4+6=16\); \(1+2+3+5+6=17\); \(1+2+4+5+6=18\); \(1+2+3+4+7=17\) | B2 | B1 for 2 or 3 correct |
| \(P(R \leq 17) = 4/462 = 2/231\) AG | A1 [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(W = 17\) | M1 | |
| \(P(W \leq 17) = \frac{2}{231}\) | ||
| Smallest \(\text{SL} = \frac{400}{231}\%\) | A1ft [2] | Allow \(\frac{4}{231}\); ft \(\frac{2}{231}\), but must be exact |
## Question 3:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Number of different rankings ${}^{11}C_5$ | M1 | Number of selections of 5 from 11 |
| $= 462$ | A1 | |
| For $R \leq 17$: $1+2+3+4+5=15$; $1+2+3+4+6=16$; $1+2+3+5+6=17$; $1+2+4+5+6=18$; $1+2+3+4+7=17$ | B2 | B1 for 2 or 3 correct |
| $P(R \leq 17) = 4/462 = 2/231$ AG | A1 **[5]** | |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $W = 17$ | M1 | |
| $P(W \leq 17) = \frac{2}{231}$ | | |
| Smallest $\text{SL} = \frac{400}{231}\%$ | A1ft **[2]** | Allow $\frac{4}{231}$; ft $\frac{2}{231}$, but must be exact |
---(i) Assuming that all rankings are equally likely, show that $\mathrm { P } ( R \leqslant 17 ) = \frac { 2 } { 231 }$.
The marks of 5 randomly chosen students from School $A$ and 6 randomly chosen students from School $B$, who took the same examination, achieving different marks, were ranked. The rankings are shown in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Rank & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\
\hline
School & $A$ & $A$ & $A$ & $B$ & $A$ & $A$ & $B$ & $B$ & $B$ & $B$ & $B$ \\
\hline
\end{tabular}
\end{center}
(ii) For a Wilcoxon rank-sum test, obtain the exact smallest significance level for which there is evidence of a difference in performance at the two schools.
\hfill \mbox{\textit{OCR S4 2010 Q3 [7]}}