| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 3×3 contingency table |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with clearly presented data in a 3×3 contingency table. Students must calculate expected frequencies, compute chi-squared contributions, compare to critical value, and interpret results. While it requires multiple computational steps and careful organization, it follows a completely routine procedure taught explicitly in S2 with no novel problem-solving or conceptual challenges beyond applying the standard algorithm. |
| Spec | 5.06a Chi-squared: contingency tables |
| Site | \multirow{2}{*}{
| ||||||
| \cline { 3 - 6 } \multicolumn{2}{|c|}{} | A | B | C | ||||
\multirow{3}{*}{
| Large | 15 | 12 | 10 | 37 | ||
| \cline { 2 - 6 } | Medium | 28 | 17 | 45 | 90 | ||
| \cline { 2 - 6 } | Small | 47 | 33 | 36 | 116 | ||
| Column totals | 90 | 62 | 91 | 243 | |||
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): no association between pebble size and site; \(H_1\): some association between pebble size and site | B1 | Must be in context. NB if \(H_0\) \(H_1\) reversed, or 'correlation' mentioned, do not award first B1 or final E1 |
| Expected values table: Large: 13.70, 9.44, 13.86; Medium: 33.33, 22.96, 33.70; Small: 42.96, 29.60, 43.44 | M1 A2 | For expected values (to 2 dp). Allow A1 for at least one row or column correct. 1d.p. can get M1A1A0. M1A2 can be implied by correct contributions/final answer |
| Contributions \(\frac{(O-E)^2}{E}\): Large: 0.1226, 0.6940, 1.0731; Medium: 0.8533, 1.5484, 3.7861; Small: 0.3793, 0.3913, 1.2744 | M1 A1 | M1 for valid attempt at \(\frac{(O-E)^2}{E}\). A1 for at least 1 row/column correct. NB These marks cannot be implied by a correct final value of \(X^2\) |
| \(X^2 = 10.12\) | M1 A1 | M1 for summation. Dependent on previous M1 |
| Refer to \(X_4^2\) | B1 | B1 for 4 degrees of freedom |
| Critical value at 5% level \(= 9.488\) | B1 | CAO for cv. Award only if no incorrect working seen |
| Result is significant | B1 | B1 ft their 'sensible' \(X^2\) and critical value. Allow reject \(H_0\). B0 if critical value of 0.711 (lower tail) or 2.776 (t distribution) used |
| There is evidence to suggest that there is some association between pebble size and site | E1 | Must be consistent with their \(X^2\). Dependent on previous B1. SC1 (to replace B1E1 if first B1B1 earned where 'significant' not stated but final statement is correct) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Site A: Contributes least to \(X^2\) showing that frequencies are as expected if there were no association. OR Contribution of 0.8533 implies that there are (slightly) fewer medium pebbles than expected. | E2,1,0 | Award E2 only if no incorrect additional comment made. Allow large/small 'as expected' or 'more than expected' and medium 'as expected' or 'less than expected' for E1 (if contribution not mentioned) |
| Site B: Contribution of 1.5484 implies that there are fewer medium pebbles than expected. | E2,1,0 | Award E2 only if no incorrect additional comment made. Allow large/small 'as expected' or 'more than expected' and medium 'less than expected' for E1 (if contribution not mentioned) |
| Site C: Contribution of 3.7861 implies that there are a lot more medium than expected. | E2,1,0 Need 'a lot more' for E2 | Award E2 only if no incorrect additional comment made. Allow large/small 'fewer than expected' and medium 'more than expected' for E1 (if contribution not mentioned) |
| NB MAX 3/6 for answers not referring to contributions (explicitly or implicitly) |
# Question 4:
## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: no association between pebble size and site; $H_1$: some association between pebble size and site | B1 | Must be in context. NB if $H_0$ $H_1$ reversed, or 'correlation' mentioned, do not award first B1 or final E1 |
| Expected values table: Large: 13.70, 9.44, 13.86; Medium: 33.33, 22.96, 33.70; Small: 42.96, 29.60, 43.44 | M1 A2 | For expected values (to 2 dp). Allow A1 for at least one row or column correct. 1d.p. can get M1A1A0. M1A2 can be implied by correct contributions/final answer |
| Contributions $\frac{(O-E)^2}{E}$: Large: 0.1226, 0.6940, 1.0731; Medium: 0.8533, 1.5484, 3.7861; Small: 0.3793, 0.3913, 1.2744 | M1 A1 | M1 for valid attempt at $\frac{(O-E)^2}{E}$. A1 for at least 1 row/column correct. NB These marks cannot be implied by a correct final value of $X^2$ |
| $X^2 = 10.12$ | M1 A1 | M1 for summation. Dependent on previous M1 |
| Refer to $X_4^2$ | B1 | B1 for 4 degrees of freedom |
| Critical value at 5% level $= 9.488$ | B1 | CAO for cv. Award only if no incorrect working seen |
| Result is significant | B1 | B1 ft their 'sensible' $X^2$ and critical value. Allow reject $H_0$. B0 if critical value of 0.711 (lower tail) or 2.776 (t distribution) used |
| There is evidence to suggest that there is some association between pebble size and site | E1 | Must be consistent with their $X^2$. Dependent on previous B1. SC1 (to replace B1E1 if first B1B1 earned where 'significant' not stated but final statement is correct) | **[12 marks]** |
## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| **Site A**: Contributes least to $X^2$ showing that frequencies are as expected if there were no association. OR Contribution of 0.8533 implies that there are (slightly) fewer medium pebbles than expected. | E2,1,0 | Award E2 only if no incorrect additional comment made. Allow large/small 'as expected' or 'more than expected' and medium 'as expected' or 'less than expected' for E1 (if contribution not mentioned) |
| **Site B**: Contribution of 1.5484 implies that there are fewer medium pebbles than expected. | E2,1,0 | Award E2 only if no incorrect additional comment made. Allow large/small 'as expected' or 'more than expected' and medium 'less than expected' for E1 (if contribution not mentioned) |
| **Site C**: Contribution of 3.7861 implies that there are **a lot** more medium than expected. | E2,1,0 Need 'a lot more' for E2 | Award E2 only if no incorrect additional comment made. Allow large/small 'fewer than expected' and medium 'more than expected' for E1 (if contribution not mentioned) |
| NB MAX 3/6 for answers not referring to contributions (explicitly or implicitly) | | | **[6 marks]** |
**TOTAL: 18 marks**4 A researcher is investigating the sizes of pebbles at various locations in a river. Three sites in the river are chosen and each pebble sampled at each site is classified as large, medium or small. The results are as follows.
\begin{center}
\begin{tabular}{ | c | l | c | c | c | c | }
\hline
\multicolumn{2}{|c|}{} & \multicolumn{3}{|c|}{Site} & \multirow{2}{*}{\begin{tabular}{ c }
Row \\
totals \\
\end{tabular}} \\
\cline { 3 - 6 }
\multicolumn{2}{|c|}{} & A & B & C & \\
\hline
\multirow{3}{*}{\begin{tabular}{ c }
Pebble \\
size \\
\end{tabular}} & Large & 15 & 12 & 10 & 37 \\
\cline { 2 - 6 }
& Medium & 28 & 17 & 45 & 90 \\
\cline { 2 - 6 }
& Small & 47 & 33 & 36 & 116 \\
\hline\hline
\multicolumn{2}{|c|}{Column totals} & 90 & 62 & 91 & 243 \\
\hline
\end{tabular}
\end{center}
(i) Carry out a test at the $5 \%$ significance level to examine whether there is any association between pebble size and site. Your working should include a table of the contributions of each cell to the test statistic.\\
(ii) By referring to each site, comment briefly on how the size of the pebbles compares with what would be expected if there were no association. You should support your answers by referring to your table of contributions.
\hfill \mbox{\textit{OCR MEI S2 2011 Q4 [18]}}