| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 2×2 contingency table |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with a 2×2 contingency table, requiring routine calculation of expected frequencies, test statistic, and comparison with critical value. While it involves multiple steps (10 marks), each step follows a well-rehearsed procedure with no conceptual challenges or novel insights required, making it slightly easier than the average A-level question. |
| Spec | 5.06a Chi-squared: contingency tables |
| \cline { 2 - 4 } \multicolumn{1}{c|}{} | Afternoon | Evening | Total |
| Win | 30 | 24 | 54 |
| Lose | 18 | 28 | 46 |
| Total | 48 | 52 | 100 |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): time of day has no effect on the outcome of a frame of snooker | B1 | \(H_0\): outcome does not depend on time of day |
| \(O_i\) | \(E_i\) | \(\ |
| 30 | 25.92 | 3.58 |
| 18 | 22.08 | 3.58 |
| 24 | 28.08 | 3.58 |
| 28 | 23.92 | 3.58 |
| 100 | 100 | |
| M1A1 | For use of Yates' correction; attempted calculation of \(\chi^2\) (even if Yates' correction not used) | |
| \(\chi^2_{5\%}(1) = 3.841\) | M1, M1, A1 | For \(\nu = 1\) and \(\chi^2\) on their value |
| \(2.07 < 3.841 \therefore\) do not reject \(H_0\) | B1B1\(\checkmark\) | |
| No evidence to suggest that the time of day has an effect on the outcome of a frame of snooker played by Syd. | A1\(\checkmark\), E1\(\checkmark\) (10 marks total) |
## Question 2:
$H_0$: time of day has no effect on the outcome of a frame of snooker | B1 | $H_0$: outcome does not depend on time of day
| $O_i$ | $E_i$ | $\|O_i - E_i\| - 0.5$ | $\chi^2$ |
|--------|--------|------------------------|----------|
| 30 | 25.92 | 3.58 | 0.4945 |
| 18 | 22.08 | 3.58 | 0.5805 |
| 24 | 28.08 | 3.58 | 0.4564 |
| 28 | 23.92 | 3.58 | 0.5358 |
| 100 | 100 | | 2.0672 |
M1A1 | For use of Yates' correction; attempted calculation of $\chi^2$ (even if Yates' correction not used)
$\chi^2_{5\%}(1) = 3.841$ | M1, M1, A1 | For $\nu = 1$ and $\chi^2$ on their value
$2.07 < 3.841 \therefore$ do not reject $H_0$ | B1B1$\checkmark$
No evidence to suggest that the time of day has an effect on the outcome of a frame of snooker played by Syd. | A1$\checkmark$, E1$\checkmark$ (10 marks total)
---2 Syd, a snooker player, believes that the outcome of any frame of snooker in which he plays may be influenced by the time of day that the frame takes place.
The results of 100 randomly selected frames of snooker, played by Syd, are recorded below.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\cline { 2 - 4 }
\multicolumn{1}{c|}{} & Afternoon & Evening & Total \\
\hline
Win & 30 & 24 & 54 \\
\hline
Lose & 18 & 28 & 46 \\
\hline
Total & 48 & 52 & 100 \\
\hline
\end{tabular}
\end{center}
Use a $\chi ^ { 2 }$ test, at the $5 \%$ level of significance, to test Syd's belief.\\
(10 marks)
\hfill \mbox{\textit{AQA S2 2005 Q2 [10]}}