| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2018 |
| Session | March |
| Marks | 9 |
| Topic | Wilcoxon tests |
| Type | Critical region or test statistic properties |
| Difficulty | Moderate -0.8 This question tests basic knowledge of Wilcoxon test conditions and properties through straightforward recall. Part (i) requires stating a standard condition (symmetry of distribution), part (ii) asks for a textbook advantage (uses magnitude information), and part (iii) involves looking up a critical value from tables for n=32 at 5% level - all routine procedures with no problem-solving or novel insight required. |
| Spec | 5.07b Sign test: and Wilcoxon signed-rank5.07c Single-sample tests |
| Answer | Marks |
|---|---|
| Distribution of times needs to be symmetric | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Uses more information | B1 | e.g. "uses magnitudes of differences" |
| Answer | Marks | Guidance |
|---|---|---|
| \(n = 32, \mu = 264\) | B1 | |
| \(\sigma^2 = 2860\) | B1 | |
| \(264 - 1.645\sqrt{2860} - 0.5\) | M1 | \(\mu - z\sigma\), allow \(\sigma^2\), ignore cc |
| A1 | Correct, with correct or no cc | |
| \(= 175.52\) | A1 | Allow 176.02 only if later rounded down to 175 |
| \(W_- \leq 175\) | A1 | Needs any letter, 175 and \(\leq\) |
| where \(W_-\) is the sum of the ranks corresponding to the negative differences | B1 | Allow \(W_-\) and "smaller of sum of positive rankings and sum of negative rankings" |
## Part (i)
Distribution of times needs to be symmetric | B1 |
## Part (ii)
Uses more information | B1 | e.g. "uses magnitudes of differences"
## Part (iii)
$n = 32, \mu = 264$ | B1 |
$\sigma^2 = 2860$ | B1 |
$264 - 1.645\sqrt{2860} - 0.5$ | M1 | $\mu - z\sigma$, allow $\sigma^2$, ignore cc
| A1 | Correct, with correct or no cc
$= 175.52$ | A1 | Allow 176.02 only if later rounded down to 175
$W_- \leq 175$ | A1 | Needs any letter, 175 and $\leq$ | Allow e.g. $W_- < 176$
where $W_-$ is the sum of the ranks corresponding to the negative differences | B1 | Allow $W_-$ and "smaller of sum of positive rankings and sum of negative rankings" | Allow any letter if properly explained4 Sheena travels to school by bus. She records the number of minutes, $T$, that her bus is late on each of 32 days. She believes that on average $T$ is greater than 5, and she carries out a significance test at the $5 \%$ level.\\
(i) State a condition needed for a Wilcoxon test to be valid in this case.
Assume now that this condition is satisfied.\\
(ii) State an advantage of using a Wilcoxon test rather than a sign test.\\
(iii) Calculate the critical region for the test, in terms of a variable which should be defined.
\hfill \mbox{\textit{OCR Further Statistics 2018 Q4 [9]}}