| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Cell combining required |
| 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, find critical value, and state conclusion—all routine S2 procedures with no conceptual complications or unusual features. Slightly above average difficulty only due to the 5×2 table size requiring careful arithmetic. |
| Spec | 5.06a Chi-squared: contingency tables |
| Against reorganisation | Not against reorganisation | ||
| \multirow{5}{*}{Age of resident} | 16-17 | 9 | 2 |
| 18-21 | 17 | 10 | |
| 22-49 | 115 | 90 | |
| 50-65 | 41 | 34 | |
| Over 65 | 3 | 4 | |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0\): no association between age and attitude to school reorganisation; \(H_1\): association between age and attitude to school reorganisation | B1 | |
| Expected values \(E_i\) attempted | M1 | E's attempted |
| At least 6 \(E_i\)'s correctly calculated | A1 | correctly (at least 6 E's) |
| \(E_i\) values: Against: \(6.262,\ 15.369,\ 116.692,\ 42.692,\ 3.985\); Not Against: \(4.738,\ 11.631,\ 88.308,\ 32.308,\ 3.015\) | ||
| Row totals: \(11, 27, 205, 75.7\ (325)\); Column totals: \(185, 140\ (325)\) | B1 | Totals correct |
| \(E_i\)'s \(< 5\): combine cells \(16\)–\(17\) and \(18\)–\(21\) also \(50\)–\(65\) and 'over \(65\)' | M1, A1 | Attempt at combining rows; Correctly |
| Final \(\dfrac{\alpha^2}{E_i}\) column attempted | ml | dep M1 |
| \(X^2 = 2.462\) | A1 | \(2.4\) to \(2.5\) |
| \(\nu = 2\) | B1 | |
| \(\chi^2_{\nu=2}(0.95) = 5.991\) | B1ft | On their \(\nu\) |
| Accept \(H_0\) | A1ft | |
| No real evidence at 5% level of significance to suggest any association between age and attitude to school reorganisation | E1ft | context |
| Total | 12 |
# Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0$: no association between age and attitude to school reorganisation; $H_1$: association between age and attitude to school reorganisation | B1 | |
| Expected values $E_i$ attempted | M1 | E's attempted |
| At least 6 $E_i$'s correctly calculated | A1 | correctly (at least 6 E's) |
| $E_i$ values: Against: $6.262,\ 15.369,\ 116.692,\ 42.692,\ 3.985$; Not Against: $4.738,\ 11.631,\ 88.308,\ 32.308,\ 3.015$ | | |
| Row totals: $11, 27, 205, 75.7\ (325)$; Column totals: $185, 140\ (325)$ | B1 | Totals correct |
| $E_i$'s $< 5$: combine cells $16$–$17$ and $18$–$21$ **also** $50$–$65$ and 'over $65$' | M1, A1 | Attempt at combining rows; Correctly |
| Final $\dfrac{\alpha^2}{E_i}$ column attempted | ml | dep M1 |
| $X^2 = 2.462$ | A1 | $2.4$ to $2.5$ |
| $\nu = 2$ | B1 | |
| $\chi^2_{\nu=2}(0.95) = 5.991$ | B1ft | On their $\nu$ |
| Accept $H_0$ | A1ft | |
| No real evidence at 5% level of significance to suggest any association between age and attitude to school reorganisation | E1ft | context |
| **Total** | **12** | |
---3 A sample survey, conducted to determine the attitudes of residents to a proposed reorganisation of local schools, gave the following results.
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{} & Against reorganisation & Not against reorganisation \\
\hline
\multirow{5}{*}{Age of resident} & 16-17 & 9 & 2 \\
\hline
& 18-21 & 17 & 10 \\
\hline
& 22-49 & 115 & 90 \\
\hline
& 50-65 & 41 & 34 \\
\hline
& Over 65 & 3 & 4 \\
\hline
\end{tabular}
\end{center}
Use a $\chi ^ { 2 }$ test, at the $5 \%$ level of significance, to determine whether there is an association between the ages of residents and their attitudes to the proposed reorganisation of local schools.
\hfill \mbox{\textit{AQA S2 2009 Q3 [12]}}