| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| 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 clearly presented contingency table data. Students must calculate expected frequencies, compute the test statistic, compare to critical value, and make a conclusion. Part (b) requires minimal interpretation. While it involves multiple calculations, it follows a routine procedure taught extensively in S2 with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{Greatest rate of income tax paid} | Age when leaving education (years) | \multirow[b]{2}{*}{Total} | ||
| 16 or less | 17 or 18 | 19 or more | ||
| Zero | 32 | 3 | 4 | 39 |
| Basic | 102 | 12 | 17 | 131 |
| Higher | 17 | 5 | 8 | 30 |
| Total | 151 | 20 | 29 | 200 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| H₀: No association between age leaving education and income tax rate; H₁: Association exists | B1 | Both hypotheses required |
| Expected frequency = \(\frac{\text{row total} \times \text{column total}}{\text{grand total}}\), e.g. \(E_{11} = \frac{39 \times 151}{200} = 29.445\) | M1 | Correct method for at least one expected frequency |
| All expected frequencies correct: Zero: 29.445, 3.9, 5.655; Basic: 98.905, 13.105, 19.0; Higher: 22.65, 3.0, 4.35 | A1 | Allow awrt 3sf |
| \(X^2 = \sum \frac{(O-E)^2}{E}\) calculated correctly | M1 | Correct formula used |
| \(X^2 \approx 7.18\) (accept range ~6.9 to 7.3) | A1 | |
| Degrees of freedom \(= (3-1)(3-1) = 4\) | B1 | |
| Critical value \(\chi^2_4(5\%) = 9.488\) | B1 | |
| Since \(7.18 < 9.488\), do not reject H₀ | M1 | Correct comparison |
| No significant evidence of association between age leaving education and income tax rate | A1 | In context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The data does not support this belief / the proportions paying higher rate do not increase consistently with age leaving education (e.g. 17/151 ≈ 11%, 5/20 = 25%, 8/29 ≈ 28% — though no significant association found) | B1 | Must refer to context; accept comment that there is no significant association so belief is not supported |
## Question 5:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| H₀: No association between age leaving education and income tax rate; H₁: Association exists | B1 | Both hypotheses required |
| Expected frequency = $\frac{\text{row total} \times \text{column total}}{\text{grand total}}$, e.g. $E_{11} = \frac{39 \times 151}{200} = 29.445$ | M1 | Correct method for at least one expected frequency |
| All expected frequencies correct: Zero: 29.445, 3.9, 5.655; Basic: 98.905, 13.105, 19.0; Higher: 22.65, 3.0, 4.35 | A1 | Allow awrt 3sf |
| $X^2 = \sum \frac{(O-E)^2}{E}$ calculated correctly | M1 | Correct formula used |
| $X^2 \approx 7.18$ (accept range ~6.9 to 7.3) | A1 | |
| Degrees of freedom $= (3-1)(3-1) = 4$ | B1 | |
| Critical value $\chi^2_4(5\%) = 9.488$ | B1 | |
| Since $7.18 < 9.488$, do not reject H₀ | M1 | Correct comparison |
| No significant evidence of association between age leaving education and income tax rate | A1 | In context |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The data does not support this belief / the proportions paying higher rate do not increase consistently with age leaving education (e.g. 17/151 ≈ 11%, 5/20 = 25%, 8/29 ≈ 28% — though no significant association found) | B1 | Must refer to context; accept comment that there is no significant association so belief is not supported |
---5 In a particular town, a survey was conducted on a sample of 200 residents aged 41 years to 50 years. The survey questioned these residents to discover the age at which they had left full-time education and the greatest rate of income tax that they were paying at the time of the survey.
The summarised data obtained from the survey are shown in the table.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multirow{2}{*}{Greatest rate of income tax paid} & \multicolumn{3}{|c|}{Age when leaving education (years)} & \multirow[b]{2}{*}{Total} \\
\hline
& 16 or less & 17 or 18 & 19 or more & \\
\hline
Zero & 32 & 3 & 4 & 39 \\
\hline
Basic & 102 & 12 & 17 & 131 \\
\hline
Higher & 17 & 5 & 8 & 30 \\
\hline
Total & 151 & 20 & 29 & 200 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use a $\chi ^ { 2 }$-test, at the $5 \%$ level of significance, to investigate whether there is an association between age when leaving education and greatest rate of income tax paid.
\item It is believed that residents of this town who had left education at a later age were more likely to be paying the higher rate of income tax. Comment on this belief.\\[0pt]
[1 mark]
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2015 Q5 [10]}}