| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| 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 standard chi-squared test of independence with straightforward calculation of expected frequencies, test statistic, and comparison to critical value. The only slight elevation above routine is identifying the largest contribution and contextualizing it, but this requires minimal additional insight beyond the mechanical procedure. |
| Spec | 5.06a Chi-squared: contingency tables |
| Good | Satisfactory | Poor | |
| Town \(A\) | 24 | 34 | 14 |
| Town \(B\) | 58 | 60 | 26 |
| Town \(C\) | 20 | 34 | 30 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(24.48\quad 30.72\quad 16.8\) | ||
| \(48.96\quad 61.44\quad 33.6\) | Find expected values (to 1 d.p.) | |
| \(28.56\quad 35.84\quad 19.6\) | M1 A1 | (lose A1 if rounded to integers) |
| \(H_0\): Grade independent of town | B1 | State (at least) null hypothesis (AEF) |
| \(\chi^2 = 0.009 + 0.350 + 0.467 + 1.669 + 0.034 + 1.719 + 2.567 + 0.094 + 5.518\) | ||
| \(= 12.4[3]\) | M1 A1 | Calculate value of \(\chi^2\) to 1 d.p. |
| \(\chi^2_{4,\,0.975} = 11.1[4]\) | B1 | Compare with consistent tabular value (to 1 d.p.) |
| Grade not independent of town | A1\(\checkmark\) | Conclusion consistent with values (AEF) |
| Town \(C\) | B1\(\checkmark\) | State town with max. contribution (on exp. values) |
| Signal poorer than expected in \(C\) | B1\(\checkmark\) | State valid comment (on exp. values) |
## Question 8:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $24.48\quad 30.72\quad 16.8$ | | |
| $48.96\quad 61.44\quad 33.6$ | | Find expected values (to 1 d.p.) |
| $28.56\quad 35.84\quad 19.6$ | M1 A1 | (lose A1 if rounded to integers) |
| $H_0$: Grade independent of town | B1 | State (at least) null hypothesis (AEF) |
| $\chi^2 = 0.009 + 0.350 + 0.467 + 1.669 + 0.034 + 1.719 + 2.567 + 0.094 + 5.518$ | | |
| $= 12.4[3]$ | M1 A1 | Calculate value of $\chi^2$ to 1 d.p. |
| $\chi^2_{4,\,0.975} = 11.1[4]$ | B1 | Compare with consistent tabular value (to 1 d.p.) |
| Grade not independent of town | A1$\checkmark$ | Conclusion consistent with values (AEF) |
| Town $C$ | B1$\checkmark$ | State town with max. contribution (on exp. values) |
| Signal poorer than expected in $C$ | B1$\checkmark$ | State valid comment (on exp. values) |
**Total: 9 marks**
---8 Residents of three towns $A , B$ and $C$ were asked to grade the reliability of their digital television signal as good, satisfactory or poor. A random sample of responses from each town is taken and the numbers in each category are given in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
& Good & Satisfactory & Poor \\
\hline
Town $A$ & 24 & 34 & 14 \\
\hline
Town $B$ & 58 & 60 & 26 \\
\hline
Town $C$ & 20 & 34 & 30 \\
\hline
\end{tabular}
\end{center}
Test, at the 2.5\% significance level, whether grade of reliability is independent of town.
Identify which town makes the greatest contribution to the test statistic and relate your answer to the context of the question.
\hfill \mbox{\textit{CAIE FP2 2012 Q8 [9]}}