| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| 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 presented data in a contingency table. Students need to calculate expected frequencies, compute the test statistic, and compare to critical values—all routine S2 procedures. Part (b) requires minimal interpretation. Slightly above average difficulty only due to the larger table size (3×4) requiring more calculations, but no conceptual challenges or novel insights needed. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | GCE Grade | \multirow[b]{2}{*}{Total} | ||||
| A | B | C | Below C | |||
| \multirow{3}{*}{Key Stage 3 Level} | 8 | 60 | 55 | 47 | 43 | 205 |
| 7 | 55 | 32 | 31 | 26 | 144 | |
| 6 | 40 | 38 | 35 | 38 | 151 | |
| Total | 155 | 125 | 113 | 107 | 500 | |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0\): No association between the performances at KS3 and GCE | B1 | |
| \(E_i\) values calculated | M1 | \(E_i\) |
| \(O_i - E_i\) values calculated | M1 | \(O_i - E_i\) |
| \(\frac{(O_i - E_i)^2}{E_i}\) values calculated | M1 | \((O_i - E_i)^2/E_i\) |
| \(\sum\) calculated | M1 | \(\sum\) |
| \(X^2 = 6.1897\) | A1 | AWFW \(6.05 - 6.35\) |
| \(\nu = 3 \times 2 = 6 \Rightarrow \chi^2_{90\%} = 10.645\) | B1B1\(\checkmark\) | on their \(\nu\) |
| Do not reject \(H_0\); No evidence to suggest an association between KS3 results and GCE grades at 10% level of significance. | E1\(\checkmark\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| More of the students achieving level 7 at KS3 gain grade A's at GCE than expected. | E1 |
# Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0$: No association between the performances at KS3 and GCE | B1 | |
| $E_i$ values calculated | M1 | $E_i$ |
| $O_i - E_i$ values calculated | M1 | $O_i - E_i$ |
| $\frac{(O_i - E_i)^2}{E_i}$ values calculated | M1 | $(O_i - E_i)^2/E_i$ |
| $\sum$ calculated | M1 | $\sum$ |
| $X^2 = 6.1897$ | A1 | AWFW $6.05 - 6.35$ |
| $\nu = 3 \times 2 = 6 \Rightarrow \chi^2_{90\%} = 10.645$ | B1B1$\checkmark$ | on their $\nu$ |
| Do not reject $H_0$; No evidence to suggest an association between KS3 results and GCE grades at 10% level of significance. | E1$\checkmark$ | |
**Total: 9 marks**
---
# Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| More of the students achieving level 7 at KS3 gain grade A's at GCE than expected. | E1 | |
**Total: 1 mark**
---7 A statistics unit is required to determine whether or not there is an association between students' performances in mathematics at Key Stage 3 and at GCE.
A survey of the results of 500 students showed the following information:
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{4}{|c|}{GCE Grade} & \multirow[b]{2}{*}{Total} \\
\hline
& & A & B & C & Below C & \\
\hline
\multirow{3}{*}{Key Stage 3 Level} & 8 & 60 & 55 & 47 & 43 & 205 \\
\hline
& 7 & 55 & 32 & 31 & 26 & 144 \\
\hline
& 6 & 40 & 38 & 35 & 38 & 151 \\
\hline
& Total & 155 & 125 & 113 & 107 & 500 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use a $\chi ^ { 2 }$ test at the $10 \%$ level of significance to determine whether there is an association between students' performances in mathematics at Key Stage 3 and at GCE.
\item Comment on the number of students who gained a grade A at GCE having gained a level 7 at Key Stage 3.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2007 Q7 [10]}}