| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Cell combining required |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with a 2×3 contingency table. Students need to calculate expected frequencies, compute the test statistic, compare to critical value, and interpret. While it requires careful arithmetic across multiple cells (11 marks suggests detailed working), the procedure is routine for S2 and requires no novel insight—just methodical application of a learned technique followed by straightforward contextual interpretation. |
| Spec | 5.06a Chi-squared: contingency tables |
| Problems during first 3 months | Problems during first year but after first 3 months | No problems during first year | Total | |
| Petrol engine | 10 | 35 | 170 | 215 |
| Diesel engine | 4 | 8 | 23 | 35 |
| Total | 14 | 43 | 193 | 250 |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| (a) | Expected values | |
| M1 | For attempt to find expected values (at least 2 correct) either before or after combining. Allow rounding to nearest integer or 1 d.p. | |
| Combine first two columns; Observed | M1 | For combining (at least 1 correct) |
| A1 | Combined Observed CAO | |
| Expected | A1 | Combined expected (both correct). Allow rounding to nearest integer or 1 d.p. |
| \(\ | (O - E)\ | - 0.5 = 3.52\) |
| \(\frac{3.52^2 + 3.52^2 + 3.52^2 + 3.52^2}{49.02 + 7.98 + 165.98 + 27.02}\) | A1 | |
| \((0.2527 + 1.5526 + 0.0746 + 0.4585) = 2.338...\) | ||
| 1 degree of freedom; CV = 2.706 | B1 | PI by CV; AWRT 2.71 |
| \(H_0\): No association between problems and engine type | B1 | Allow "problems independent of engine type" but not reverse or "they are independent" or simply "no association" |
| \(2.34 < 2.706\) (so accept \(H_0\)) | A1 dep | Comparison stated or diagram equivalent; Dep on A1 for ts and B1 for CV |
| There is no significant evidence of an association between mechanical problems and type of engine. | E1 dep | OE Conclusion in context. Dep on previous A1; Dep on A1 dep for ts and B1 for CV |
| Total: 11 | ||
| (b) | Dependent on a conclusion of no association. | E1 |
| No evidence that diesels have more problems | E1 dep | Or similar dep on previous E1 |
| Go with her preference or buy either. | ||
| Total: 2 | ||
| TOTAL FOR Q5: 13 |
| Part | Answer/Working | Marks | Guidance |
|------|---|---|---|
| (a) | **Expected values** | | |
| | | M1 | For attempt to find expected values (at least 2 correct) either before or after combining. Allow rounding to nearest integer or 1 d.p. |
| | **Combine first two columns; Observed** | M1 | For combining (at least 1 correct) |
| | | A1 | Combined Observed CAO |
| | **Expected** | A1 | Combined expected (both correct). Allow rounding to nearest integer or 1 d.p. |
| | $\|(O - E)\| - 0.5 = 3.52$ | M1 | Attempt including Yates (at least 1 correct); AWRT 2.34 (Implies first 6 marks) |
| | $\frac{3.52^2 + 3.52^2 + 3.52^2 + 3.52^2}{49.02 + 7.98 + 165.98 + 27.02}$ | A1 | |
| | $(0.2527 + 1.5526 + 0.0746 + 0.4585) = 2.338...$ | | |
| | 1 degree of freedom; CV = 2.706 | B1 | PI by CV; AWRT 2.71 |
| | $H_0$: No association between problems and engine type | B1 | Allow "problems independent of engine type" but not reverse or "they are independent" or simply "no association" |
| | $2.34 < 2.706$ (so accept $H_0$) | A1 dep | Comparison stated or diagram equivalent; Dep on A1 for ts and B1 for CV |
| | There is no significant evidence of an association between mechanical problems and type of engine. | E1 dep | OE Conclusion in context. Dep on previous A1; Dep on A1 dep for ts and B1 for CV |
| | **Total: 11** | | |
| (b) | Dependent on a conclusion of **no association**. | E1 | Even from incorrect working; Use of result, however any additional suggestion that diesels may have more problems then E0 |
| | No evidence that diesels have more problems | E1 dep | Or similar dep on previous E1 |
| | Go with her preference or buy either. | | |
| | **Total: 2** | | |
| | **TOTAL FOR Q5: 13** | | |
**Notes on Q5:**
- No Yates: Can score M1, M1, A1, A1, M0, A0, B1, B1 (for 2 d.o.f. giving 4.605), B1, A0, E0 = 7 out of 11 ($\chi^2 = 3.05$)
- No combining: Can score M1, M0, A0, A0, M0, A0, B1, B1 (for 2 d.o.f. giving 4.605), B1, A0, E0 = 4 out of 11 ($\chi^2 = 3.92$)
- Just 2 cells combined (1.96 and 6.02) can score M1 M1 and B1 for hypotheses
---A car manufacturer keeps a record of how many of the new cars that it has sold experience mechanical problems during the first year. The manufacturer also records whether the cars have a petrol engine or a diesel engine. Data for a random sample of 250 cars are shown in the table.
\begin{tabular}{|l|c|c|c|c|}
\hline
& Problems during first 3 months & Problems during first year but after first 3 months & No problems during first year & Total \\
\hline
Petrol engine & 10 & 35 & 170 & 215 \\
\hline
Diesel engine & 4 & 8 & 23 & 35 \\
\hline
Total & 14 & 43 & 193 & 250 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\alph*)]
\item Use a $\chi^2$-test to investigate, at the 10% significance level, whether there is an association between the mechanical problems experienced by a new car from this manufacturer and the type of engine. [11 marks]
\item Arisa is planning to buy a new car from this manufacturer. She would prefer to buy a car with a diesel engine, but a friend has told her that cars with diesel engines experience more mechanical problems.
Based on your answer to part (a), state, with a reason, the advice that you would give to Arisa. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2016 Q5 [13]}}