| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| 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 a 2×3 contingency table requiring routine calculation of expected frequencies, test statistic, and comparison with critical value. Part (b) adds minimal difficulty by asking for interpretation of results. The question is slightly above average difficulty due to being Further Maths content, but the execution is entirely procedural with no novel insight required. |
| Spec | 5.06a Chi-squared: contingency tables |
| Company | Poor | Satisfactory | Good |
| \(P\) | 18 | 43 | 64 |
| \(Q\) | 22 | 22 | 31 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| There is sufficient evidence to suggest that the quality of brushes is not independent of company. | A1 | Correct work only, conclusion in context with level of uncertainty in language. Not 'prove'. Accept 'enough evidence'. Condone 'some evidence'. Do not accept 'there is insufficient…'. |
| Total: 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P\) produces better quality brushes (than \(Q\)) or \(Q\) produces worse quality brushes (than \(P\)); \(P\) has fewer poor quality brushes than expected AND \(Q\) has more poor quality brushes than expected | B1 | Condone \(P\) is better (than \(Q\)). |
| Total: 1 |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| There is sufficient evidence to suggest that the quality of brushes is not independent of company. | A1 | Correct work only, conclusion in context with level of uncertainty in language. Not 'prove'. Accept 'enough evidence'. Condone 'some evidence'. Do not accept 'there is insufficient…'. |
| **Total: 7** | | |
---
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P$ produces better quality brushes (than $Q$) or $Q$ produces worse quality brushes (than $P$); $P$ has fewer poor quality brushes than expected AND $Q$ has more poor quality brushes than expected | B1 | Condone $P$ is better (than $Q$). |
| **Total: 1** | | |
---5 Two companies, $P$ and $Q$, produce a certain type of paint brush. An independent examiner rates the quality of the brushes produced as poor, satisfactory or good. He takes a random sample of brushes from each company. The examiner's ratings are summarised in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
Company & Poor & Satisfactory & Good \\
\hline
$P$ & 18 & 43 & 64 \\
\hline
$Q$ & 22 & 22 & 31 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Test, at the $5 \%$ significance level, whether quality of brushes is independent of company.
\item Compare the quality of the brushes produced by the two companies.
\end{enumerate}
\hfill \mbox{\textit{CAIE Further Paper 4 2024 Q5 [8]}}