| Exam Board | AQA |
|---|---|
| Module | Further Paper 3 Statistics (Further Paper 3 Statistics) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Cell combining required |
| Difficulty | Moderate -0.5 This is a straightforward chi-squared test of independence question requiring standard recall of hypotheses, the rule about expected frequencies <5, the test statistic formula, and table lookup. All parts are routine bookwork with no problem-solving or novel insight required, making it easier than average but not trivial since it's a Further Maths topic. |
| Spec | 5.06a Chi-squared: contingency tables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(H_0\): There is no association between town and number of speeding offences per year. \(H_1\): There is an association between town and number of speeding offences per year. | B1 (AO2.5) | Variables need to be stated in at least the null hypothesis |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The column with expected frequency less than 5 needs to be merged with another column | E1 (AO3.5c) | Explains merging of column/row with expected frequency less than 5 |
| This applies to both observed and expected frequencies | E1 (AO3.5c) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sum \frac{(O-E)^2}{E}\) | B1 (AO1.2) | States correct test statistic |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\chi^2\) cv for 25 df \(= 44.314\); \(45.22 > 44.314\) | R1 (AO3.5a) | Compares correct critical value with test statistic |
| Reject \(H_0\) | E1 (AO2.2b) | |
| Some evidence to suggest there is an association between town and number of speeding offences per year | E1F (AO3.2a) | Conclusion must not be definite; FT incorrect acceptance of \(H_0\) if stated |
# Question 6(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $H_0$: There is no association between town and number of speeding offences per year. $H_1$: There is an association between town and number of speeding offences per year. | B1 (AO2.5) | Variables need to be stated in at least the null hypothesis |
---
# Question 6(b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The column with expected frequency less than 5 needs to be merged with another column | E1 (AO3.5c) | Explains merging of column/row with expected frequency less than 5 |
| This applies to both observed and expected frequencies | E1 (AO3.5c) | |
---
# Question 6(b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sum \frac{(O-E)^2}{E}$ | B1 (AO1.2) | States correct test statistic |
---
# Question 6(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\chi^2$ cv for 25 df $= 44.314$; $45.22 > 44.314$ | R1 (AO3.5a) | Compares correct critical value with test statistic |
| Reject $H_0$ | E1 (AO2.2b) | |
| Some evidence to suggest there is an association between town and number of speeding offences per year | E1F (AO3.2a) | Conclusion must not be definite; FT incorrect acceptance of $H_0$ if stated |
---6 Danai is investigating the number of speeding offences in different towns in a country.
She carries out a hypothesis test to test for association between town and number of speeding offences per year.
6
\begin{enumerate}[label=(\alph*)]
\item State the hypotheses for this test.
6
\item The observed frequencies, $O$, have been collected and the expected frequencies, $E$, have been calculated in an $n \times m$ contingency table, where $n > 3$ and $m > 3$
One of the values of $E$ is less than 5
6 (b) (i) Explain what steps Danai should take before calculating the test statistic.\\
6 (b) (ii) State an expression for the test statistic Danai should calculate.\\
6
\item Danai correctly calculates the value of the test statistic to be 45.22
The number of degrees of freedom for the test is 25\\
Determine the outcome of Danai's test, using the $1 \%$ level of significance.
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 3 Statistics 2021 Q6 [7]}}