| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2013 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Interpret association after test |
| Difficulty | Standard +0.3 This is a straightforward chi-squared test of independence with clear data presentation. Students must calculate expected frequencies, compute the test statistic with contributions from each cell, compare to critical value at 10% level, and make basic interpretations. While it requires multiple computational steps, it follows a standard procedure taught explicitly in S2 with no conceptual surprises or novel problem-solving required. |
| Spec | 5.06a Chi-squared: contingency tables |
| Artist preferred | |||||
| \cline { 3 - 6 } \multicolumn{2}{|c|}{} | Monet | Renoir | Degas | Cézanne | |
| \multirow{2}{*}{Sex} | Male | 8 | 25 | 18 | 19 |
| \cline { 2 - 6 } | Female | 18 | 35 | 10 | 17 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): no association between sex and artist preferred | B1 | For both hypotheses in context |
| \(H_1\): some association between sex and artist preferred | ||
| Expected values table: Male: 12.13, 28, 13.07, 16.8; Female: 13.87, 32, 14.93, 19.2 | M1 A2 | For expected values (to 2 dp where appropriate); allow A1 for at least one row or column correct |
| Contributions \((O-E)^2/E\) table: Male: 1.4081, 0.3214, 1.8626, 0.2881; Female: 1.2321, 0.2813, 1.6298, 0.2521 | M1 A2 | For valid attempt at \((O-E)^2/E\); for all correct (to 2 dp) presented in a table or clear list; allow A1 for at least one row or column correct; NB: these three marks cannot be implied by a correct final value of \(X^2\) |
| \(X^2 = 7.28\) | B1 | Allow 7.27; www |
| Refer to \(\chi_3^2\) | B1 | For 3 deg of f |
| Critical value at 10% level \(= 6.251\) | B1 | CAO for cv; no FT from here if wrong or omitted, unless \(p\)-value used instead; B1 for \(p\)-value \(= 0.0636\) |
| Result is significant | B1 | FT their \(X^2\) |
| There is evidence to suggest that there is some association between sex and artist preferred | E1 | For correct (FT their \(X^2\)), non-assertive conclusion, in context |
| NB if \(H_0\ H_1\) reversed, or 'correlation' mentioned, do not award first B1 or final E1 | ||
| [12] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Monet: More females and fewer males than expected prefer Monet, as indicated by large contribution(s) (of 1.4081 and 1.2321) | E1* E1dep* | FT their table of contributions; NB MAX 3/6 for answers not referring to contributions (explicitly or implicitly) |
| Renoir: Preferences are much as expected, as indicated by small contributions | E1 | |
| Degas: Fewer females and more males than expected prefer Degas, as indicated by large contribution(s) (of 1.8626 and 1.6298) | E1* depE1* | |
| Cézanne: Preferences are much as expected, as indicated by small contributions | E1 | SC1 Renoir and Cézanne have correct comments for both but without referring to contributions |
| [6] |
# Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: no association between sex and artist preferred | B1 | For both hypotheses in context |
| $H_1$: some association between sex and artist preferred | | |
| Expected values table: Male: 12.13, 28, 13.07, 16.8; Female: 13.87, 32, 14.93, 19.2 | M1 A2 | For expected values (to 2 dp where appropriate); allow A1 for at least one row or column correct |
| Contributions $(O-E)^2/E$ table: Male: 1.4081, 0.3214, 1.8626, 0.2881; Female: 1.2321, 0.2813, 1.6298, 0.2521 | M1 A2 | For valid attempt at $(O-E)^2/E$; for all correct (to 2 dp) presented in a table or clear list; allow A1 for at least one row or column correct; NB: these three marks cannot be implied by a correct final value of $X^2$ |
| $X^2 = 7.28$ | B1 | Allow 7.27; www |
| Refer to $\chi_3^2$ | B1 | For 3 deg of f |
| Critical value at 10% level $= 6.251$ | B1 | CAO for cv; no FT from here if wrong or omitted, unless $p$-value used instead; B1 for $p$-value $= 0.0636$ |
| Result is significant | B1 | FT their $X^2$ |
| There is evidence to **suggest** that there is some association between sex and artist preferred | E1 | For correct (FT their $X^2$), non-assertive conclusion, in context |
| NB if $H_0\ H_1$ reversed, or 'correlation' mentioned, do not award first B1 or final E1 | | |
| **[12]** | | |
---
# Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Monet: **More females and fewer males than expected** prefer Monet, as indicated by **large contribution(s)** (of 1.4081 and 1.2321) | E1* E1dep* | FT their table of contributions; NB MAX 3/6 for answers not referring to contributions (explicitly or implicitly) |
| Renoir: Preferences are much **as expected**, as indicated by **small contributions** | E1 | |
| Degas: **Fewer females and more males than expected** prefer Degas, as indicated by **large contribution(s)** (of 1.8626 and 1.6298) | E1* depE1* | |
| Cézanne: Preferences are much **as expected**, as indicated by **small contributions** | E1 | SC1 Renoir and Cézanne have correct comments for both but without referring to contributions |
| **[6]** | | |4 An art gallery is holding an exhibition. A random sample of 150 visitors to the exhibition is selected. The visitors are asked which of four artists they prefer. Their preferences, classified according to whether the visitor is female or male, are given in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
\multicolumn{2}{|c|}{} & \multicolumn{4}{|c|}{Artist preferred} \\
\cline { 3 - 6 }
\multicolumn{2}{|c|}{} & Monet & Renoir & Degas & Cézanne \\
\hline
\multirow{2}{*}{Sex} & Male & 8 & 25 & 18 & 19 \\
\cline { 2 - 6 }
& Female & 18 & 35 & 10 & 17 \\
\hline
\end{tabular}
\end{center}
(i) Carry out a test at the $10 \%$ significance level to examine whether there is any association between artist preferred and sex of visitor. Your working should include a table showing the contributions of each cell to the test statistic.\\
(ii) For each artist, comment briefly on how the preferences of each sex compare with what would be expected if there were no association.
\hfill \mbox{\textit{OCR MEI S2 2013 Q4 [18]}}