| Exam Board | OCR MEI |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2009 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon matched-pairs signed-rank test |
| Difficulty | Standard +0.3 This is a straightforward application of the Wilcoxon signed-rank test with clear paired data. Students must calculate differences, rank them (ignoring one zero difference), sum ranks, and compare to tables. The systematic sampling part (i) is basic sampling theory. While it requires careful arithmetic, it follows a standard textbook procedure with no novel insight needed, making it slightly easier than average. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc5.07b Sign test: and Wilcoxon signed-rank |
| Staff member | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Uniform A | 4.2 | 2.6 | 10.0 | 9.0 | 8.2 | 2.8 | 5.0 | 7.4 | 2.8 | 6.8 | 10.0 | 9.8 |
| Uniform B | 5.0 | 5.2 | 1.4 | 2.8 | 2.2 | 6.4 | 7.4 | 7.8 | 6.8 | 1.2 | 3.4 | 7.6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| She needs a list of all staff | E1 | |
| with no cycles in the list | E1 | |
| All staff equally likely to be chosen if she chooses a random start between 1 and 10 | E1 | |
| then chooses every \(10^{\text{th}}\) | E1 | |
| Not simple random sampling since not all samples are possible | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Nothing is known about the background population | E1 | Any reference to unknown distribution or "non-parametric" situation |
| ...of differences between the scores | E1 | Any reference to pairing/differences |
| \(H_0: m = 0\), \(H_1: m \neq 0\) | B1 | Both hypotheses. Hypotheses in words only must include "population" |
| where \(m\) is the population median difference for the scores | B1 | For adequate verbal definition |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Differences and ranks table: Diff: \(-0.8, -2.6, 8.6, 6.2, 6.0, -3.6, -2.4, -0.4, -4.0, 5.6, 6.6, 2.2\); Rank: \(2, 5, 12, 10, 9, 6, 4, 1, 7, 8, 11, 3\) | M1 | For differences. ZERO in this section if differences not used |
| M1 | For ranks | |
| A1 | ft from here if ranks wrong | |
| \(W_- = 1 + 2 + 4 + 5 + 6 + 7 = 25\) | B1 | (or \(W_+ = 3 + 8 + 9 + 10 + 11 + 12 = 53\)) |
| Refer to tables of Wilcoxon paired (single sample) statistic for \(n = 12\) | M1 | No ft from here if wrong |
| Lower (or upper if 53 used) \(2\frac{1}{2}\)% tail is 13 (or 65 if 53 used) | A1 | i.e. a 2-tail test. No ft from here if wrong |
| Result is not significant | A1 | ft only c's test statistic |
| No evidence to suggest a preference for one of the uniforms | A1 | ft only c's test statistic |
# Question 3:
## Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| She needs a list of all staff | E1 | |
| with no cycles in the list | E1 | |
| All staff equally likely to be chosen if she chooses a random start between 1 and 10 | E1 | |
| then chooses every $10^{\text{th}}$ | E1 | |
| Not simple random sampling since not all samples are possible | E1 | |
## Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Nothing is known about the background population | E1 | Any reference to unknown distribution or "non-parametric" situation |
| ...of differences between the scores | E1 | Any reference to pairing/differences |
| $H_0: m = 0$, $H_1: m \neq 0$ | B1 | Both hypotheses. Hypotheses in words only must include "population" |
| where $m$ is the population median difference for the scores | B1 | For adequate verbal definition |
## Part (iii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differences and ranks table: Diff: $-0.8, -2.6, 8.6, 6.2, 6.0, -3.6, -2.4, -0.4, -4.0, 5.6, 6.6, 2.2$; Rank: $2, 5, 12, 10, 9, 6, 4, 1, 7, 8, 11, 3$ | M1 | For differences. ZERO in this section if differences not used |
| | M1 | For ranks |
| | A1 | ft from here if ranks wrong |
| $W_- = 1 + 2 + 4 + 5 + 6 + 7 = 25$ | B1 | (or $W_+ = 3 + 8 + 9 + 10 + 11 + 12 = 53$) |
| Refer to tables of Wilcoxon paired (single sample) statistic for $n = 12$ | M1 | No ft from here if wrong |
| Lower (or upper if 53 used) $2\frac{1}{2}$% tail is 13 (or 65 if 53 used) | A1 | i.e. a 2-tail test. No ft from here if wrong |
| Result is not significant | A1 | ft only c's test statistic |
| No evidence to suggest a preference for one of the uniforms | A1 | ft only c's test statistic |
---3 A company which employs 600 staff wishes to improve its image by introducing new uniforms for the staff to wear. The human resources manager would like to obtain the views of the staff. She decides to do this by means of a systematic sample of $10 \%$ of the staff.\\
(i) How should she go about obtaining such a sample, ensuring that all members of staff are equally likely to be selected? Explain whether this constitutes a simple random sample.
At a later stage in the process, the choice of uniform has been reduced to two possibilities. Twelve members of staff are selected to take part in deciding which of the two uniforms to adopt. Each of the twelve assesses each uniform for comfort, appearance and practicality, giving it a total score out of 10. The scores are as follows.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Staff member & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
Uniform A & 4.2 & 2.6 & 10.0 & 9.0 & 8.2 & 2.8 & 5.0 & 7.4 & 2.8 & 6.8 & 10.0 & 9.8 \\
\hline
Uniform B & 5.0 & 5.2 & 1.4 & 2.8 & 2.2 & 6.4 & 7.4 & 7.8 & 6.8 & 1.2 & 3.4 & 7.6 \\
\hline
\end{tabular}
\end{center}
A Wilcoxon signed rank test is to be used to decide whether there is any evidence of a preference for one of the uniforms.\\
(ii) Explain why this test is appropriate in these circumstances and state the hypotheses that should be used.\\
(iii) Carry out the test at the $5 \%$ significance level.
\hfill \mbox{\textit{OCR MEI S3 2009 Q3 [17]}}