| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 2×3 contingency table |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with a 2×3 contingency table. Students need to calculate expected frequencies, compute the test statistic, find critical value from tables, and state conclusion. While it requires multiple computational steps, it's a routine application of a well-practiced procedure with no conceptual challenges or novel insights required. Slightly easier than average due to straightforward setup and clear structure. |
| Spec | 5.06a Chi-squared: contingency tables |
| Area 1 | Area 2 | Area 3 | |
| Local bus service | 73 | 36 | 30 |
| Road surfaces | 47 | 44 | 20 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(66.72\ \ 44.48\ \ 27.80\) | M1 A1 | Find expected values (to 1 dp); lose A1 if rounded to integers |
| \(53.28\ \ 35.52\ \ 22.20\) | ||
| \(H_0\): No association between them | B1 | State (at least) null hypothesis (A.E.F.) |
| \(\chi^2 = 5.36 \pm 0.03\) | M1 A1 | Calculate value of \(\chi^2\) |
| \(\chi^2_{2,\,0.95} = 5.991\) | B1 | Compare with consistent tabular value (to 2 dp) |
| No association | A1\(\sqrt{}\) | Conclusion consistent with values (A.E.F.) |
## Question 6:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $66.72\ \ 44.48\ \ 27.80$ | M1 A1 | Find expected values (to 1 dp); lose A1 if rounded to integers |
| $53.28\ \ 35.52\ \ 22.20$ | | |
| $H_0$: No association between them | B1 | State (at least) null hypothesis (A.E.F.) |
| $\chi^2 = 5.36 \pm 0.03$ | M1 A1 | Calculate value of $\chi^2$ |
| $\chi^2_{2,\,0.95} = 5.991$ | B1 | Compare with consistent tabular value (to 2 dp) |
| No association | A1$\sqrt{}$ | Conclusion consistent with values (A.E.F.) |
**Total: [7]**
---6 A random sample of residents in a town took part in a survey. They were asked whether they would prefer the local council to spend money on improving the local bus service or on improving the quality of road surfaces. The responses are shown in the following table, classified according to the area of the town in which the residents live.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
& Area 1 & Area 2 & Area 3 \\
\hline
Local bus service & 73 & 36 & 30 \\
\hline
Road surfaces & 47 & 44 & 20 \\
\hline
\end{tabular}
\end{center}
Using a $5 \%$ significance level, test whether there is an association between the area lived in and preference for improving the local bus service or improving the quality of road surfaces.
\hfill \mbox{\textit{CAIE FP2 2011 Q6 [7]}}