| Exam Board | OCR MEI |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2011 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon signed-rank test (single sample) |
| Difficulty | Standard +0.3 This is a straightforward application of stratified sampling calculations and a standard Wilcoxon signed-rank test. Part (a) requires simple proportional allocation (multiply 2000 by each stratum's proportion), and part (b) follows the routine procedure of finding differences, ranking absolute values, summing ranks, and comparing to critical values. No novel insight or complex reasoning required—slightly easier than average due to clear structure and standard methodology. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc5.06a Chi-squared: contingency tables5.07b Sign test: and Wilcoxon signed-rank |
| Tax band | A - B | C - D | E - F | G - H |
| Number of households | 32298 | 33211 | 9739 | 4120 |
| Local authority | A | B | C | D | E | F | G | H | I | J |
| 1999 | 9.60 | 8.40 | 8.67 | 9.32 | 9.89 | 9.35 | 7.91 | 8.08 | 9.61 | 8.55 |
| 2009 | 8.94 | 8.42 | 7.87 | 8.41 | 10.17 | 10.11 | 8.31 | 9.76 | 9.54 | 9.67 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| There are identifiable subgroups or strata that might exhibit different characteristics | E1 | |
| Each stratum is randomly sampled | E1 | |
| Use it to obtain a representative sample | E1 | |
| Can get information on the individual strata | E1 | Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| For each stratum \(\ldots \times \frac{2000}{79368}\) giving 813.9, 836.9, 245.4, 103.8 so 814, 837, 245, 104 | M1, A1 | All correct. Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| The population (or underlying distribution) is assumed to be symmetrical about its median | E2 | E2, 1, 0. Award E1 for 2 out of 3 of the key features. Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0: m = 0\), \(H_1: m \neq 0\) where \(m\) is the population median difference for the percentages | B1, B1 | Both hypotheses in words, only must include "population". For adequate verbal definition |
| Differences table with ranks as shown | M1 | For differences. ZERO (out of 8) in this section if paired differences not used |
| Ranks calculated | M1, A1 | For ranks; ft from here if ranks wrong |
| \(W_- = 2 + 5 + 7 + 8 = 22\) | B1 | (or \(W_+ = 1+3+4+6+9+10 = 33\)) |
| Refer to tables of Wilcoxon paired/single sample statistic for \(n = 10\) | M1 | No ft from here if wrong |
| Lower (or upper if 33 used) 5% tail is 10 (or 45 if 33 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 change in spending on average | A1 | ft only c's test statistic. Conclusion in context to include "on average" o.e. Total: 10 |
# Question 2:
## Part (a)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| There are identifiable subgroups or strata that might exhibit different characteristics | E1 | |
| Each stratum is randomly sampled | E1 | |
| Use it to obtain a representative sample | E1 | |
| Can get information on the individual strata | E1 | Total: 4 |
## Part (a)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| For each stratum $\ldots \times \frac{2000}{79368}$ giving 813.9, 836.9, 245.4, 103.8 so 814, 837, 245, 104 | M1, A1 | All correct. Total: 2 |
## Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| The population (or underlying distribution) is assumed to be symmetrical about its median | E2 | E2, 1, 0. Award E1 for 2 out of 3 of the key features. Total: 2 |
## Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: m = 0$, $H_1: m \neq 0$ where $m$ is the population median difference for the percentages | B1, B1 | Both hypotheses in words, only must include "population". For adequate verbal definition |
| Differences table with ranks as shown | M1 | For differences. ZERO (out of 8) in this section if paired differences not used |
| Ranks calculated | M1, A1 | For ranks; ft from here if ranks wrong |
| $W_- = 2 + 5 + 7 + 8 = 22$ | B1 | (or $W_+ = 1+3+4+6+9+10 = 33$) |
| Refer to tables of Wilcoxon paired/single sample statistic for $n = 10$ | M1 | No ft from here if wrong |
| Lower (or upper if 33 used) 5% tail is 10 (or 45 if 33 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 change in spending on average | A1 | ft only c's test statistic. Conclusion in context to include "on average" o.e. Total: 10 |
---2
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item What is stratified sampling? Why would it be used?
\item A local authority official wishes to conduct a survey of households in the borough. He decides to select a stratified sample of 2000 households using Council Tax property bands as the strata. At the time of the survey there are 79368 households in the borough. The table shows the numbers of households in the different tax bands.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
Tax band & A - B & C - D & E - F & G - H \\
\hline
Number of households & 32298 & 33211 & 9739 & 4120 \\
\hline
\end{tabular}
\end{center}
Calculate the number of households that the official should choose from each stratum in order to obtain his sample of 2000 households so that each stratum is represented proportionally.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item What assumption needs to be made when using a Wilcoxon single sample test?
\item As part of an investigation into trends in local authority spending, one of the categories of expenditure considered was 'Highways and the Environment'. For a random sample of 10 local authorities, the percentages of their total expenditure spent on Highways and the Environment in 1999 and then in 2009 are shown in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | }
\hline
Local authority & A & B & C & D & E & F & G & H & I & J \\
\hline
1999 & 9.60 & 8.40 & 8.67 & 9.32 & 9.89 & 9.35 & 7.91 & 8.08 & 9.61 & 8.55 \\
\hline
2009 & 8.94 & 8.42 & 7.87 & 8.41 & 10.17 & 10.11 & 8.31 & 9.76 & 9.54 & 9.67 \\
\hline
\end{tabular}
\end{center}
Use a Wilcoxon test, with a significance level of $10 \%$, to determine whether there appears to be any change to the average percentage of total expenditure spent on Highways and the Environment between 1999 and 2009.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI S3 2011 Q2 [18]}}