| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2022 |
| Session | November |
| Marks | 7 |
| 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 a 3×3 contingency table. Students must calculate expected frequencies, compute the test statistic, find critical value from tables, and state conclusion. While it requires careful arithmetic and proper hypothesis testing procedure, it follows a completely routine template with no conceptual challenges or novel elements—slightly easier than average due to its mechanical nature. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | Region | ||||
| A | B | C | Total | ||
| \multirow{3}{*}{Performance} | Above target | 62 | 41 | 44 | 147 |
| On target | 102 | 94 | 95 | 291 | |
| Below target | 56 | 45 | 61 | 162 | |
| Total | 220 | 180 | 200 | 600 | |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(H_0\): performance is independent of region; \(H_1\): performance is not independent of region | B1 | |
| 62 (53.9), 41 (44.1), 44 (49); 102 (106.7), 94 (87.3), 95 (97); 56 (59.4), 45 (48.6), 61 (54) | M1 A1 | Calculate expected frequencies, allow 1 error |
| Test stat \(= 1.217 + 0.218 + 0.510 + 0.207 + 0.514 + 0.041 + 0.195 + 0.267 + 0.907\) | M1 | Calculate test statistic |
| 4.08 | A1 | SCB2 for 4.08 if totally unsupported. SCM1A1 B1 for 4.08 supported only by expected frequencies |
| Tabular value, \(4df = 7.779\); \(1.08 < 7.779\); Accept \(H_0\) | M1 | Compare with 7.779 and conclusion |
| Insufficient evidence that performance is not independent of region | A1 | Correct conclusion in context, following correct work, level of uncertainty in language. 'Prove' gives A0 |
| 7 |
## Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $H_0$: performance is independent of region; $H_1$: performance is not independent of region | B1 | |
| 62 (53.9), 41 (44.1), 44 (49); 102 (106.7), 94 (87.3), 95 (97); 56 (59.4), 45 (48.6), 61 (54) | M1 A1 | Calculate expected frequencies, allow 1 error |
| Test stat $= 1.217 + 0.218 + 0.510 + 0.207 + 0.514 + 0.041 + 0.195 + 0.267 + 0.907$ | M1 | Calculate test statistic |
| 4.08 | A1 | SCB2 for 4.08 if totally unsupported. SCM1A1 B1 for 4.08 supported only by expected frequencies |
| Tabular value, $4df = 7.779$; $1.08 < 7.779$; Accept $H_0$ | M1 | Compare with 7.779 and conclusion |
| Insufficient evidence that performance is not independent of region | A1 | Correct conclusion in context, following correct work, level of uncertainty in language. 'Prove' gives A0 |
| | **7** | |
---2 In the colleges in three regions of a particular country, students are given individual targets to achieve. Their performance is measured against their individual target and graded as 'above target', 'on target' or 'below target'. For a random sample of students from each of the three regions, the observed frequencies are summarised in the following table.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{3}{|c|}{Region} & \\
\hline
& & A & B & C & Total \\
\hline
\multirow{3}{*}{Performance} & Above target & 62 & 41 & 44 & 147 \\
\hline
& On target & 102 & 94 & 95 & 291 \\
\hline
& Below target & 56 & 45 & 61 & 162 \\
\hline
& Total & 220 & 180 & 200 & 600 \\
\hline
\end{tabular}
\end{center}
Test, at the 10\% significance level, whether performance is independent of region.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2022 Q2 [7]}}