| Exam Board | CAIE |
|---|---|
| Module | Further Paper 4 (Further Paper 4) |
| Year | 2024 |
| Session | June |
| 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 clearly presented contingency table data. Students must calculate expected frequencies, compute the test statistic using the formula, find degrees of freedom (2×2=4), and compare to critical value. While it requires careful arithmetic with 9 cells, it follows a completely routine procedure taught in Further Statistics with no conceptual challenges or novel elements. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | Bus company | ||||
| \(A\) | \(B\) | \(C\) | Total | ||
| \multirow{3}{*}{Arrival} | Early | 22 | 22 | 10 | 54 |
| On time | 30 | 52 | 42 | 124 | |
| Late | 28 | 26 | 18 | 72 | |
| Total | 80 | 100 | 70 | 250 | |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Expected values table: Arrived early: 22 (17.28), 22 (21.60), 10 (15.12); Arrived on time: 30 (39.68), 52 (49.60), 42 (34.72); Arrived late: 28 (23.04), 26 (28.80), 18 (20.16) | M1 A1 | Calculate expected values, at least 4 correct to 3sf. These must be seen. A1 for all correct to at least 3sf. |
| \(\chi^2\) contributions: 1.2893, 0.00741, 1.7338; 2.3615, 0.1161, 1.5265; 1.0678, 0.2722, 0.2314 | M1 | At least 4 correct values (to 2sf) or expressions. May be implied by \(8.600 - 8.615\). |
| Test statistic \(= 8.61\) | A1 | \(8.600 - 8.615\). Unsupported \(8.600 - 8.615\) scores M0A0 M1A1. |
| \(H_0\): Reliability of buses is independent of bus company; \(H_1\): Reliability of buses is not independent of/dependent on bus company | B1 | Condone 'No association/relationship between…' 'Association/relationship between…' |
| \(8.61 < 9.488\), Accept \(H_0\) | M1 | '8.61' from an attempt at chi-squared, 9.488 must be correct. ft conclusion from their 8.61. Condone 'Reject \(H_1\)'. 'Accept \(H_0\)' can be implied by a conclusion in context consistent with their 8.61. |
| There is insufficient evidence to support/suggest that reliability of buses is dependent/not independent of bus company | A1 | Correct work only, ignoring their hypotheses, conclusion in context with level of uncertainty in language. Not 'prove' or equivalent. Condone 'no sufficient' 'not enough'. Do not accept 'there is sufficient evidence to suggest….' |
| Total: 7 |
# Question 3:
| Answer | Mark | Guidance |
|--------|------|----------|
| Expected values table: Arrived early: 22 **(17.28)**, 22 **(21.60)**, 10 **(15.12)**; Arrived on time: 30 **(39.68)**, 52 **(49.60)**, 42 **(34.72)**; Arrived late: 28 **(23.04)**, 26 **(28.80)**, 18 **(20.16)** | M1 A1 | Calculate expected values, at least 4 correct to 3sf. These must be seen. A1 for all correct to at least 3sf. |
| $\chi^2$ contributions: 1.2893, 0.00741, 1.7338; 2.3615, 0.1161, 1.5265; 1.0678, 0.2722, 0.2314 | M1 | At least 4 correct values (to 2sf) or expressions. May be implied by $8.600 - 8.615$. |
| Test statistic $= 8.61$ | A1 | $8.600 - 8.615$. Unsupported $8.600 - 8.615$ scores M0A0 M1A1. |
| $H_0$: **Reliability** of buses **is independent of** bus company; $H_1$: **Reliability** of buses **is not independent of/dependent on** bus company | B1 | Condone 'No association/relationship between…' 'Association/relationship between…' |
| $8.61 < 9.488$, Accept $H_0$ | M1 | '8.61' from an attempt at chi-squared, 9.488 must be correct. ft conclusion from their 8.61. Condone 'Reject $H_1$'. 'Accept $H_0$' can be implied by a conclusion in context consistent with their 8.61. |
| There is insufficient evidence to support/suggest that reliability of buses is dependent/not independent of bus company | A1 | Correct work only, ignoring their hypotheses, conclusion in context with level of uncertainty in language. Not 'prove' or equivalent. Condone 'no sufficient' 'not enough'. Do not accept 'there is sufficient evidence to suggest….' |
| **Total: 7** | | |
---3 There are three bus companies in a city. The council is investigating whether the buses reliably arrive at their destination on time. The results from random samples of buses from each company 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|}{Bus company} & \\
\hline
& & $A$ & $B$ & $C$ & Total \\
\hline
\multirow{3}{*}{Arrival} & Early & 22 & 22 & 10 & 54 \\
\hline
& On time & 30 & 52 & 42 & 124 \\
\hline
& Late & 28 & 26 & 18 & 72 \\
\hline
& Total & 80 & 100 & 70 & 250 \\
\hline
\end{tabular}
\end{center}
Test, at the $5 \%$ significance level, whether the reliability of buses is independent of bus company.\\
\hfill \mbox{\textit{CAIE Further Paper 4 2024 Q3 [7]}}