| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | January |
| Marks | 10 |
| 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 clearly structured data requiring routine calculation of expected frequencies, test statistic, and comparison with critical value. The interpretation in part (b) is straightforward. Slightly above average difficulty due to the 3×2 contingency table requiring multiple calculations, but follows a completely standard procedure taught in S2 with no conceptual challenges or novel insights required. |
| Spec | 5.06a Chi-squared: contingency tables |
| Age | Pass | Fail |
| \(\mathbf { 1 7 } - \mathbf { 1 8 }\) | 28 | 20 |
| \(\mathbf { 1 9 } - \mathbf { 3 0 }\) | 2 | 14 |
| \(\mathbf { 3 1 } - \mathbf { 3 9 }\) | 12 | 33 |
| \(\mathbf { 4 0 } - \mathbf { 6 0 }\) | 6 | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0\): no association between age and first time performance in driving test; \(H_1\): association between age and first time performance | B1 | |
| Expected values calculated correctly (19.2, 28.8, 6.4, 9.6, 18.0, 27.0, 4.4, 6.6) | M1A1 | E's attempted correctly |
| \(\frac{(O-E)^2}{E}\) column: 4.0333, 3.0250, 0.8643, 2.6889, 2.0167, 0.5762; sum = 13.20 | M1A1, m1A1 | Attempt at combining correctly; final column attempted; for \(X^2\) correct |
| \(\nu = 2 \Rightarrow \chi^2(2) = 9.210\) | B1ft | on \(\nu=2\) or \(\nu=3\) only |
| Reject \(H_0\); evidence to support Julie's belief at 1% level of significance | E1ft |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| More students than expected in age group 17–18 pass their test first time | E1 | Fewer than expected fail |
## Question 4(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0$: no association between age and first time performance in driving test; $H_1$: association between age and first time performance | B1 | |
| Expected values calculated correctly (19.2, 28.8, 6.4, 9.6, 18.0, 27.0, 4.4, 6.6) | M1A1 | E's attempted correctly |
| $\frac{(O-E)^2}{E}$ column: 4.0333, 3.0250, 0.8643, 2.6889, 2.0167, 0.5762; sum = 13.20 | M1A1, m1A1 | Attempt at combining correctly; final column attempted; for $X^2$ correct |
| $\nu = 2 \Rightarrow \chi^2(2) = 9.210$ | B1ft | on $\nu=2$ or $\nu=3$ only |
| Reject $H_0$; evidence to support Julie's belief at 1% level of significance | E1ft | |
## Question 4(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| More students than expected in age group 17–18 pass their test first time | E1 | Fewer than expected fail |4 Julie, a driving instructor, believes that the first-time performances of her students in their driving tests are associated with their ages.
Julie's records of her students' first-time performances in their driving tests are shown in the table.
\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
Age & Pass & Fail \\
\hline
$\mathbf { 1 7 } - \mathbf { 1 8 }$ & 28 & 20 \\
\hline
$\mathbf { 1 9 } - \mathbf { 3 0 }$ & 2 & 14 \\
\hline
$\mathbf { 3 1 } - \mathbf { 3 9 }$ & 12 & 33 \\
\hline
$\mathbf { 4 0 } - \mathbf { 6 0 }$ & 6 & 5 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use a $\chi ^ { 2 }$ test at the $1 \%$ level of significance to investigate Julie's belief.
\item Interpret your result in part (a) as it relates to the 17-18 age group.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2010 Q4 [10]}}