| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2018 |
| Session | December |
| Marks | 15 |
| Topic | Wilcoxon tests |
| Type | Wilcoxon signed-rank test (single sample) |
| Difficulty | Standard +0.3 This is a straightforward application of standard hypothesis testing procedures with clear step-by-step parts. Part (a) is a routine one-sample z-test, part (b)(i) requires recalling a standard comparison between tests, part (b)(ii) is a textbook Wilcoxon signed-rank test with small sample size (tables provided), and part (c) asks for a standard conceptual explanation about normality assumptions. All techniques are direct applications with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.07b Sign test: and Wilcoxon signed-rank5.07e Test medians |
| Answer | Marks |
|---|---|
| B2, M1, A1, A1, M1ft, A1ft | One error, e.g. no or wrong letter, ≠, etc : B1; Find sample mean; Correct z, p or CV; Correct comparison; Correct first conclusion; Context, not too definite (e.g. not "international athletes' reaction times are shorter"); Needs correct method, like-with-like; ft on their z, p or CV; ft on their z, p or CV |
| Answer | Marks |
|---|---|
| B1 | (no additional guidance) |
| Answer | Marks |
|---|---|
| B1, M1, A1, A1, A1, A1ft | Same as in (i) but different letter or "median" stated; Find differences from 700, rank and consider signs; For \(W_-\) or \(W_+\); For both, and T correct; Correct CV; In context, not too definite; FT on their T |
| Answer | Marks |
|---|---|
| B1 | Not "one is more accurate" |
## (a)
$H_0: \mu = 700$
$H_1: \mu < 700$ where $\mu$ is the mean reaction time for all international athletes
$\bar{x} = 607$
$z = -1.822$ or $p = 0.0342$ or CV = 616.05...
$z < -1.645$ or $p < 0.05$ or $607 < \text{CV}$
Reject $H_0$
Significant evidence that mean reaction times of international athletes are shorter
| B2, M1, A1, A1, M1ft, A1ft | One error, e.g. no or wrong letter, ≠, etc : B1; Find sample mean; Correct z, p or CV; Correct comparison; Correct first conclusion; Context, not too definite (e.g. not "international athletes' reaction times are shorter"); Needs correct method, like-with-like; ft on their z, p or CV; ft on their z, p or CV |
---
# Question 6 (b)
## (i)
Uses more information (e.g. magnitudes of differences)
| B1 | (no additional guidance) |
## (ii)
$H_0: m = 700, H_1: m < 700$
where $m$ is the median reaction time for all international athletes
Differences: 2 –69 –160 14 –125 –220
Ranks: +1 –3 –5 +2 –4 –6
$W_- = 18$
$W_+ = 3$ so $T = 3$
$n = 6, \text{CV} = 2$
Do not reject $H_0$. Insufficient evidence that median reaction times of international athletes are shorter
| B1, M1, A1, A1, A1, A1ft | Same as in (i) but different letter or "median" stated; Find differences from 700, rank and consider signs; For $W_-$ or $W_+$; For both, and T correct; Correct CV; In context, not too definite; FT on their T |
## (c)
They use different assumptions
| B1 | Not "one is more accurate" |
---6 The reaction times, in milliseconds, of all adult males in a standard experiment have a symmetrical distribution with mean and median both equal to 700 and standard deviation 125.
The reaction times of a random sample of 6 international athletes are measured and the results are as follows:\\
$\begin{array} { l l l l l l } 702 & 631 & 540 & 714 & 575 & 480 \end{array}$
It is required to test whether international athletes have a mean reaction time which is less than 700.
\begin{enumerate}[label=(\alph*)]
\item Assume first that the reaction times of international athletes have the distribution $\mathrm { N } \left( \mu , 125 ^ { 2 } \right)$.
Test at the $5 \%$ significance level whether $\mu < 700$.
\item Now assume only that the distribution of the data is symmetrical, but not necessarily normal.
\begin{enumerate}[label=(\roman*)]
\item State with a reason why a Wilcoxon test is preferable to a sign test.
\item Use an appropriate Wilcoxon test at the $5 \%$ significance level to test whether the median reaction time of international athletes is less than 700 .
\end{enumerate}\item Explain why the significance tests in part (a) and part (b)(ii) could produce different results.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2018 Q6 [15]}}