| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2023 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Larger contingency table (4+ categories) |
| Difficulty | Moderate -0.3 This is a standard chi-squared test of independence with a 2×4 contingency table. It requires routine calculation of expected frequencies, test statistic, and comparison with critical value—all textbook procedures with no novel insight needed. Slightly easier than average due to its straightforward setup and clear structure. |
| Spec | 5.06a Chi-squared: contingency tables |
| \cline { 2 - 5 } \multicolumn{1}{c|}{} | North Street | East Street | South Street | West Street |
| Approve | 33 | 54 | 42 | 26 |
| Disapprove | 19 | 39 | 28 | 9 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0\): opinion is independent of street; \(H_1\): opinion is not independent of street | B1 | |
| Expected values: 33 32.24 \ | 54 57.66 \ | 42 43.4 \ |
| (As above, all 6 expected values correct) | A1 | 6 correct values or expressions |
| \(0.0179 + 0.2323 + 0.0452 + 0.8521 + 0.0292 + 0.3790 + 0.0737 + 1.3902\) | M1 | Correct to 3 d.p.; at least 2 correct values or expressions |
| \(3.02\) | A1 | SC B1 3.02 following M1M0; SC B2 3.02 following M0M0 |
| Tabular value, 3 degrees of freedom \(7.815\): \(3.02 < 7.815\), accept \(H_0\) | M1 | Allow 'not significant' |
| Insufficient evidence to suggest that opinion depends on street | A1 | CWO. Correct conclusion in context, following correct work, level of uncertainty in language. A0 if hypotheses wrong way round or missing |
## Question 2:
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0$: opinion is independent of street; $H_1$: opinion is not independent of street | B1 | |
| Expected values: 33 **32.24** \| 54 **57.66** \| 42 **43.4** \| 26 **21.7** \| 155; 19 **19.76** \| 39 **35.34** \| 28 **26.6** \| 9 **13.3** \| 95; 52 \| 93 \| 70 \| 35 \| 250 | M1 | At least 2 correct values or expressions |
| (As above, all 6 expected values correct) | A1 | 6 correct values or expressions |
| $0.0179 + 0.2323 + 0.0452 + 0.8521 + 0.0292 + 0.3790 + 0.0737 + 1.3902$ | M1 | Correct to 3 d.p.; at least 2 correct values or expressions |
| $3.02$ | A1 | SC B1 3.02 following M1M0; SC B2 3.02 following M0M0 |
| Tabular value, 3 degrees of freedom $7.815$: $3.02 < 7.815$, accept $H_0$ | M1 | Allow 'not significant' |
| Insufficient evidence to suggest that opinion depends on street | A1 | CWO. Correct conclusion in context, following correct work, level of uncertainty in language. A0 if hypotheses wrong way round or missing |
---2 A town council has published its plans for redeveloping the town centre and residents are being asked whether they approve or disapprove. A random sample of 250 responses has been selected from residents in the four main streets in the town: North, East, South and West Streets. The results are shown in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\cline { 2 - 5 }
\multicolumn{1}{c|}{} & North Street & East Street & South Street & West Street \\
\hline
Approve & 33 & 54 & 42 & 26 \\
\hline
Disapprove & 19 & 39 & 28 & 9 \\
\hline
\end{tabular}
\end{center}
Test, at the $5 \%$ significance level, whether the opinions of the residents are independent of the streets on which they live.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2023 Q2 [7]}}