| Exam Board | OCR MEI |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2013 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Wilcoxon tests |
| Type | Wilcoxon matched-pairs signed-rank test |
| Difficulty | Moderate -0.3 This is a straightforward application of two standard statistical tests (Wilcoxon signed-rank and chi-squared goodness of fit) with clear instructions and tabulated data. Part (a) requires calculating differences, ranking them, and comparing to critical values from tables. Part (b) is a routine chi-squared test with equal expected frequencies. Both are textbook exercises requiring careful execution but no problem-solving insight or novel approaches, making this slightly easier than average. |
| Spec | 2.01a Population and sample: terminology5.06b Fit prescribed distribution: chi-squared test5.07b Sign test: and Wilcoxon signed-rank |
| Project | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Examiner A | 58 | 37 | 72 | 78 | 67 | 77 | 62 | 41 | 80 | 60 | 65 | 70 |
| Examiner B | 73 | 47 | 74 | 71 | 78 | 96 | 54 | 27 | 97 | 73 | 60 | 66 |
| Digit | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Frequency | 6 | 8 | 11 | 14 | 12 | 9 | 15 | 5 | 14 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Number all the projects to be marked. (Sampling frame.) | E1 | Do not award if candidate subsequently describes a different method of sampling (e.g. systematic sampling). |
| Use a form of random number generator to select the projects in the sample until 12 projects have been selected. | E1 | Condone absence of 12. |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0: m = 0 \quad H_1: m \neq 0\) where \(m\) is the population median difference between the examiners' marks. | This is given in the question. | |
| Differences: \(15, 10, 2, -7, 11, 19, -8, -14, 17, 13, -5, -4\) | M1 | For differences. ZERO (out of 8) in this section if differences not used. |
| Ranks: \(10, 6, 1, 4, 7, 12, 5, 9, 11, 8, 3, 2\) | M1 | For ranks. |
| A1 | ft from here if ranks wrong. | |
| \(W_- = 2+3+4+5+9 = 23\) | B1 | (or \(W_+ = 1+6+7+8+10+11+12 = 55\)) |
| Refer to tables of Wilcoxon paired (/single sample) statistic for \(n = 12\). | M1 | No ft from here if wrong. |
| Lower (or upper if 55 used) 5% tail is 17 (or 61 if 55 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. |
| Insufficient evidence to suggest a difference in the marks awarded, on average. | A1 | ft only c's test statistic. Conclusion in context to include "average" o.e. |
| [8] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): The random number function is performing as it should. \(H_1\): The random number function is not performing as it should. | B1 | Both hypotheses. Must be the right way round. Allow use of the uniform distribution/model. Do not accept "data fit model" oe. |
| All expected frequencies are 10 | B1 | |
| \(X^2 = 1.6 + 0.4 + 0.1 + 1.6 + 0.4 + 0.1 + 2.5 + 2.5 + 1.6 + 1.6\) | M1 | Calculation of \(X^2\). |
| \(= 12.4\) | A1 | c.a.o. |
| Refer to \(\chi^2_9\) | M1 | Allow correct df (\(= \text{cells} - 1\)) from wrongly grouped table and ft. Otherwise, no ft if wrong. \(P(X^2 > 12.4) = 0.1916\). |
| Upper 10% point is 14.68. | A1 | No ft from here if wrong. |
| Not significant. | A1 | ft only c's test statistic. |
| Insufficient evidence to suggest that the random number function is not performing as it should. | A1 | ft only c's test statistic. Conclusion in context. Allow in terms of the uniform distribution/model. Do not accept "data fit model" oe. |
| [8] |
# Question 4:
## Part (a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Number all the projects to be marked. (Sampling frame.) | E1 | Do not award if candidate subsequently describes a different method of sampling (e.g. systematic sampling). |
| Use a form of random number generator to select the projects in the sample until 12 projects have been selected. | E1 | Condone absence of 12. |
| **[2]** | | |
## Part (a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0: m = 0 \quad H_1: m \neq 0$ where $m$ is the population median difference between the examiners' marks. | | This is given in the question. |
| Differences: $15, 10, 2, -7, 11, 19, -8, -14, 17, 13, -5, -4$ | M1 | For differences. ZERO (out of 8) in this section if differences not used. |
| Ranks: $10, 6, 1, 4, 7, 12, 5, 9, 11, 8, 3, 2$ | M1 | For ranks. |
| | A1 | ft from here if ranks wrong. |
| $W_- = 2+3+4+5+9 = 23$ | B1 | (or $W_+ = 1+6+7+8+10+11+12 = 55$) |
| Refer to tables of Wilcoxon paired (/single sample) statistic for $n = 12$. | M1 | No ft from here if wrong. |
| Lower (or upper if 55 used) 5% tail is 17 (or 61 if 55 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. |
| Insufficient evidence to suggest a difference in the marks awarded, on average. | A1 | ft only c's test statistic. Conclusion in context to include "average" o.e. |
| **[8]** | | |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: The random number function is performing as it should. $H_1$: The random number function is not performing as it should. | B1 | Both hypotheses. Must be the right way round. Allow use of the uniform distribution/model. Do not accept "data fit model" oe. |
| All expected frequencies are 10 | B1 | |
| $X^2 = 1.6 + 0.4 + 0.1 + 1.6 + 0.4 + 0.1 + 2.5 + 2.5 + 1.6 + 1.6$ | M1 | Calculation of $X^2$. |
| $= 12.4$ | A1 | c.a.o. |
| Refer to $\chi^2_9$ | M1 | Allow correct df ($= \text{cells} - 1$) from wrongly grouped table and ft. Otherwise, no ft if wrong. $P(X^2 > 12.4) = 0.1916$. |
| Upper 10% point is 14.68. | A1 | No ft from here if wrong. |
| Not significant. | A1 | ft only c's test statistic. |
| Insufficient evidence to suggest that the random number function is not performing as it should. | A1 | ft only c's test statistic. Conclusion in context. Allow in terms of the uniform distribution/model. Do not accept "data fit model" oe. |
| | **[8]** | |4
\begin{enumerate}[label=(\alph*)]
\item At a college, two examiners are responsible for marking, independently, the students' projects. Each examiner awards a mark out of 100 to each project. There is some concern that the examiners' marks do not agree, on average. Consequently a random sample of 12 projects is selected and the marks awarded to them are compared.
\begin{enumerate}[label=(\roman*)]
\item Describe how a random sample of projects should be chosen.
\item The marks given for the projects in the sample are as follows.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | c | c | }
\hline
Project & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
Examiner A & 58 & 37 & 72 & 78 & 67 & 77 & 62 & 41 & 80 & 60 & 65 & 70 \\
\hline
Examiner B & 73 & 47 & 74 & 71 & 78 & 96 & 54 & 27 & 97 & 73 & 60 & 66 \\
\hline
\end{tabular}
\end{center}
Carry out a test at the $10 \%$ level of significance of the hypotheses $\mathrm { H } _ { 0 } : m = 0 , \mathrm { H } _ { 1 } : m \neq 0$, where $m$ is the population median difference.
\end{enumerate}\item A calculator has a built-in random number function which can be used to generate a list of random digits. If it functions correctly then each digit is equally likely to be generated. When it was used to generate 100 random digits, the frequencies of the digits were as follows.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | c | }
\hline
Digit & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
Frequency & 6 & 8 & 11 & 14 & 12 & 9 & 15 & 5 & 14 & 6 \\
\hline
\end{tabular}
\end{center}
Use a goodness of fit test, with a significance level of $10 \%$, to investigate whether the random number function is generating digits with equal probability.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S3 2013 Q4 [18]}}