| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 11 |
| 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 for independence (or homogeneity) with a 2×2 contingency table. Students need to state hypotheses, calculate expected frequencies, compute the chi-squared statistic, find the critical value at 10% significance, and draw a conclusion. While it requires multiple steps and careful calculation, it follows a routine procedure taught in S3 with no conceptual surprises, making it slightly easier than average for an A-level statistics question. |
| Spec | 5.06a Chi-squared: contingency tables |
| English | History | |
| Highfield School | 32 | 14 |
| Rowntree School | 48 | 26 |
| Answer | Marks |
|---|---|
| expected freq. Highfield/English \(= \frac{80+46}{120} = 30.67\) | M1 A1 |
| giving expected freqs | M1 A1 |
| Answer | Marks |
|---|---|
| \(H_0 :\) no difference in proportions at the two schools | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| O | E | (O−E) |
| 32 | 30.67 | 1.33 |
| 14 | 15.33 | −1.33 |
| 48 | 49.33 | −1.33 |
| 26 | 24.67 | 1.33 |
| \(\therefore \Sigma \frac{(O-E)^2}{E} = 0.2807\) | M1 A2 | |
| \(\nu = 1, \chi^2_{\text{crit}}(10\%) = 2.705\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| there is no evidence of a difference in proportions at the two schools | A1 | (11) |
expected freq. Highfield/English $= \frac{80+46}{120} = 30.67$ | M1 A1 |
giving expected freqs | M1 A1 |
30.67 15.33
49.33 24.67
$H_0 :$ no difference in proportions at the two schools | B1 |
$H_1 :$ there is a difference in proportions at the two schools
| O | E | (O−E) | $\frac{(O-E)^2}{E}$ |
|---|---|-------|-----|
| 32 | 30.67 | 1.33 | 0.0577 |
| 14 | 15.33 | −1.33 | 0.1154 |
| 48 | 49.33 | −1.33 | 0.0359 |
| 26 | 24.67 | 1.33 | 0.0717 |
$\therefore \Sigma \frac{(O-E)^2}{E} = 0.2807$ | M1 A2 |
$\nu = 1, \chi^2_{\text{crit}}(10\%) = 2.705$ | M1 A1 |
0.2807 < 2.705 $\therefore$ not significant
there is no evidence of a difference in proportions at the two schools | A1 | (11)Two schools in the same town advertise at the same time for new heads of English and History departments. The number of applicants for each post are shown in the table below.
\begin{center}
\begin{tabular}{|l|c|c|}
\hline
& English & History \\
\hline
Highfield School & 32 & 14 \\
\hline
Rowntree School & 48 & 26 \\
\hline
\end{tabular}
\end{center}
Stating your hypotheses clearly, test at the 10\% level of significance whether or not there is evidence of the proportion of applicants for each job being different in the two schools. [11 marks]
\hfill \mbox{\textit{Edexcel S3 Q6 [11]}}