| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| 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 labeled contingency table data. Students must calculate expected frequencies, compute the test statistic, compare to critical value, and make contextual comments. While it requires multiple steps, it follows a routine procedure taught in S2 with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | Class of degree | |||||
| 1 | 2(i) | 2(ii) | 3 | Total | ||
| \multirow{2}{*}{A-level grade} | A | 20 | 36 | 22 | 2 | 80 |
| B | 9 | 55 | 48 | 8 | 120 | |
| Total | 29 | 91 | 70 | 10 | 200 | |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): No association between A level grade and class of degree | B1 | At least \(H_0\) correct |
| \(H_1\): Association between A level grade and class of degree | ||
| M1 | For \(E_i\)'s attempted | |
| M1 | For combining attempted | |
| M1 | For final column attempted | |
| A1 | (awrt 13.5) | |
| \(\nu = 2\) | B1 | [\(\nu = 3\) with \(\chi^2 = 11.345\) (B0B1fi)] |
| \(\chi_{1\%}^2(2) = 9.210\) | B1 | |
| Reject \(H_0\) | A1 | Dep on B1 M1M1M1 B1B1, not A1 |
| Fiona's belief justified | E1 | 9 |
| Answer | Marks | Guidance |
|---|---|---|
| Fewer than expected gained a Class 1 degree having gained grade B in A-level Mathematics. | E1 | |
| More than expected gained a Class 2(ii) degree having gained grade B in A-level Mathematics. | E1 | 2 |
| A: fewer than expected | ||
| B: as expected | ||
| C: more than expected | ||
| D: more or similar than expected | ||
| E: more than expected | ||
| 11 |
### 6(a)
| $H_0$: No association between A level grade and class of degree | B1 | At least $H_0$ correct |
| --- | --- | --- |
| $H_1$: Association between A level grade and class of degree | | |
| | M1 | For $E_i$'s attempted |
| | M1 | For combining attempted |
| | M1 | For final column attempted |
| | A1 | (awrt 13.5) |
| $\nu = 2$ | B1 | [$\nu = 3$ with $\chi^2 = 11.345$ (B0B1fi)] |
| $\chi_{1\%}^2(2) = 9.210$ | B1 | |
| **Reject $H_0$** | A1 | Dep on B1 M1M1M1 B1B1, not A1 |
| **Fiona's belief justified** | E1 | 9 | Dep on B1 M1M1M1 B1B1, not A1 |
### 6(b)
| Fewer than expected gained a Class 1 degree having gained grade B in A-level Mathematics. | E1 | |
| --- | --- | --- |
| **More than expected gained a Class 2(ii) degree having gained grade B in A-level Mathematics.** | E1 | 2 | correct comments (see below) |
| | | |
| | | A: fewer than expected |
| | | B: as expected |
| | | C: more than expected |
| | | D: more or similar than expected |
| | | E: more than expected |
| | | **11** | |6 Fiona, a lecturer in a school of engineering, believes that there is an association between the class of degree obtained by her students and the grades that they had achieved in A-level Mathematics.
In order to investigate her belief, she collected the relevant data on the performances of a random sample of 200 recent graduates who had achieved grades A or B in A-level Mathematics. These data are tabulated below.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{4}{|c|}{Class of degree} & \\
\hline
& & 1 & 2(i) & 2(ii) & 3 & Total \\
\hline
\multirow{2}{*}{A-level grade} & A & 20 & 36 & 22 & 2 & 80 \\
\hline
& B & 9 & 55 & 48 & 8 & 120 \\
\hline
& Total & 29 & 91 & 70 & 10 & 200 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Conduct a $\chi ^ { 2 }$ test, at the $1 \%$ level of significance, to determine whether Fiona's belief is justified.
\item Make two comments on the degree performance of those students in this sample who achieved a grade B in A-level Mathematics.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2012 Q6 [11]}}