| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2008 |
| Session | January |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Larger contingency table (4+ categories) |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with clearly structured data and straightforward hypotheses. Part (a) requires routine calculation of expected frequencies and chi-squared contributions following a standard template (12 marks reflects length rather than difficulty). Part (b) is a basic one-sample z-test with given parameters. Both parts are textbook applications requiring no novel insight, though slightly above average difficulty due to the computational work and table construction required. |
| Spec | 5.06a Chi-squared: contingency tables |
| Mathematics only | English only | Both | Neither | Row totals | |
| Male | 38 | 19 | 6 | 32 | 95 |
| Female | 42 | 55 | 9 | 49 | 155 |
| Column totals | 80 | 74 | 15 | 81 | 250 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_1\): some association between sex and subject; | B1 | 1 |
| OBS | Maths | English |
| Male | 38 | 19 |
| Female | 42 | 55 |
| Col sum | 80 | 74 |
| EXP | Maths | English |
| Male | 30.40 | 28.12 |
| Female | 49.60 | 45.88 |
| Col sum | 80 | 74 |
| CONT | Maths | English |
| Male | 1.900 | 2.958 |
| Female | 1.165 | 1.813 |
| Answer | Marks | Guidance |
|---|---|---|
| NB If \(H_0, H_1\) reversed, or 'correlation' mentioned, do not award first B1 or final E1 | M1 A2 for expected values (allow A1 for at least one row or column correct), M1 for valid attempt at \((O-E)^2/E\), A1 NB These M1 A1 marks cannot be implied by a correct final value of \(X^2\), M1 for summation, A1 cao for \(X^2\), B1 for 3 deg of f, B1 CAO for cv, B1, E1 | 7, 4 |
| Answer | Marks | Guidance |
|---|---|---|
| There is insufficient evidence to conclude that the mean score is increased by the new teaching method. | B1 for both correct, B1 for definition of \(\mu\), M1, A1 cao, B1 for 1.282, M1 for comparison, A1 for conclusion in words and in context | 7 |
**(i)** $H_0$: no association between sex and subject;
$H_1$: some association between sex and subject; | B1 | 1
| OBS | Maths | English | Both | Neither | Row sum |
|-----|-------|---------|------|---------|----------|
| Male | 38 | 19 | 6 | 32 | 95 |
| Female | 42 | 55 | 9 | 49 | 155 |
| Col sum | 80 | 74 | 15 | 81 | 250 |
| EXP | Maths | English | Both | Neither | Row sum |
|-----|-------|---------|------|---------|----------|
| Male | 30.40 | 28.12 | 5.70 | 30.78 | 95 |
| Female | 49.60 | 45.88 | 9.30 | 50.22 | 155 |
| Col sum | 80 | 74 | 15 | 81 | 250 |
| CONT | Maths | English | Both | Neither |
|------|-------|---------|------|---------|
| Male | 1.900 | 2.958 | 0.016 | 0.048 |
| Female | 1.165 | 1.813 | 0.010 | 0.030 |
$X^2 = 7.94$
Refer to $\chi_3^2$;
Critical value at 5% level = 7.815
Result is significant
There is evidence to suggest that there is some association between sex and subject choice.
NB If $H_0, H_1$ reversed, or 'correlation' mentioned, do not award first B1 or final E1 | M1 A2 for expected values (allow A1 for at least one row or column correct), M1 for valid attempt at $(O-E)^2/E$, A1 NB These M1 A1 marks cannot be implied by a correct final value of $X^2$, M1 for summation, A1 cao for $X^2$, B1 for 3 deg of f, B1 CAO for cv, B1, E1 | 7, 4
**(ii)** $H_0: \mu = 67.4$; $H_1: \mu > 67.4$
Where $\mu$ denotes the mean score of the population of students taught with the new method.
Test statistic $= \frac{68.3 - 67.4}{8.9/\sqrt{12}} = \frac{0.9}{2.57} = 0.35$
10% level 1 tailed critical value of $z = 1.282$
0.35 < 1.282 so not significant.
There is insufficient evidence to reject $H_0$
There is insufficient evidence to conclude that the mean score is increased by the new teaching method. | B1 for both correct, B1 for definition of $\mu$, M1, A1 cao, B1 for 1.282, M1 for comparison, A1 for conclusion in words and in context | 7
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**Total for Question 4: 19**4
\begin{enumerate}[label=(\alph*)]
\item A researcher believes that there may be some association between a student's sex and choice of certain subjects at A-level. A random sample of 250 A -level students is selected. The table below shows, for each sex, how many study either or both of the two subjects, Mathematics and English.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
& Mathematics only & English only & Both & Neither & Row totals \\
\hline
Male & 38 & 19 & 6 & 32 & 95 \\
\hline
Female & 42 & 55 & 9 & 49 & 155 \\
\hline
Column totals & 80 & 74 & 15 & 81 & 250 \\
\hline
\end{tabular}
\end{center}
Carry out a test at the $5 \%$ significance level to examine whether there is any association between a student's sex and choice of subjects. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic. [12]
\item Over a long period it has been determined that the mean score of students in a particular English module is 67.4 and the standard deviation is 8.9. A new teaching method is introduced with the aim of improving the results. A random sample of 12 students taught by the new method is selected. Their mean score is found to be 68.3. Carry out a test at the $10 \%$ level to investigate whether the new method appears to have been successful. State carefully your null and alternative hypotheses. You should assume that the scores are Normally distributed and that the standard deviation is unchanged.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI S2 2008 Q4 [19]}}