| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 2×3 contingency table |
| Difficulty | Standard +0.3 This is a standard chi-squared test for independence/homogeneity with a 3×2 contingency table. Students must state hypotheses, calculate expected frequencies, compute the test statistic, find critical value, and conclude. While it requires multiple steps and careful calculation, it follows a routine procedure taught explicitly in S3 with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.06a Chi-squared: contingency tables |
| Minor | Serious | |
| 8 a.m. – 6 p.m. | 45 | 11 |
| 6 p.m. – 2 a.m. | 49 | 22 |
| 2 a.m. – 8 a.m. | 14 | 7 |
| Answer | Marks |
|---|---|
| Expected frequencies: 8am-6pm/minor \(= \frac{108-56}{148} = 40.86\); 6pm-2am/minor \(= \frac{108-74}{148} = 51.81\) | M1 A2 |
| Answer | Marks | Guidance |
|---|---|---|
| 40.86 | 15.14 | |
| 51.81 | 19.19 | |
| 15.33 | 5.67 | |
| A1 | ||
| Hypothesis setup: \(H_0\): proportion of serious injuries independent of time; \(H_1\): proportion of serious injuries varies with time | B1 | |
| Chi-squared calculation: \(\sum \frac{(O-E)^2}{E} = 2.543\) | M1 A2 | |
| Critical value and conclusion: \(\nu = 2\); \(\chi^2_{crit}(5\%) = 5.991\); \(2.543 < 5.991\) \(\therefore\) not significant; there is no evidence of prop'n of serious injuries varying with time | M1 A1 A1 | (11 marks total) |
**Expected frequencies:** 8am-6pm/minor $= \frac{108-56}{148} = 40.86$; 6pm-2am/minor $= \frac{108-74}{148} = 51.81$ | M1 A2 |
**Expected frequency table:**
| | | |
|---|---|---|
| 40.86 | 15.14 | |
| 51.81 | 19.19 | |
| 15.33 | 5.67 | |
| A1 |
**Hypothesis setup:** $H_0$: proportion of serious injuries independent of time; $H_1$: proportion of serious injuries varies with time | B1 |
**Chi-squared calculation:** $\sum \frac{(O-E)^2}{E} = 2.543$ | M1 A2 |
**Critical value and conclusion:** $\nu = 2$; $\chi^2_{crit}(5\%) = 5.991$; $2.543 < 5.991$ $\therefore$ not significant; there is no evidence of prop'n of serious injuries varying with time | M1 A1 A1 | (11 marks total)
---A hospital administrator is assessing staffing needs for its Accident and Emergency Department at different times of day. The administrator already has data on the number of admissions at different times of day but needs to know if the proportion of the cases that are serious remains constant.
Staff are asked to assess whether each person arriving at Accident and Emergency has a "minor" or "serious" problem and the results for three different time periods are shown below.
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
& Minor & Serious \\
\hline
8 a.m. – 6 p.m. & 45 & 11 \\
\hline
6 p.m. – 2 a.m. & 49 & 22 \\
\hline
2 a.m. – 8 a.m. & 14 & 7 \\
\hline
\end{tabular}
\end{center}
Stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence of the proportion of serious injuries being different at different times of day. [11]
\hfill \mbox{\textit{Edexcel S3 Q4 [11]}}