| Exam Board | AQA |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Percentages given, table construction required |
| Difficulty | Moderate -0.3 This is a standard chi-squared test of independence with straightforward setup: converting percentages to frequencies (simple multiplication), calculating expected values, and applying the test procedure. The only minor challenge is the percentage-to-frequency conversion, but this is routine A-level work with clear structure and no conceptual surprises. |
| Spec | 5.06a Chi-squared: contingency tables |
| Age |
|
| ||||
| \(\mathbf { 2 2 - } \mathbf { 3 4 }\) | 17.5 | 40.0 | ||||
| \(\mathbf { 3 5 - } \mathbf { 3 9 }\) | 60.0 | 45.0 | ||||
| \(\mathbf { 4 0 - } \mathbf { 5 9 }\) | 22.5 | 15.0 |
| Answer | Marks | Guidance |
|---|---|---|
| A | B | Total |
| 22-34 | 21 | 32 |
| 35-39 | 72 | 36 |
| 40-59 | 27 | 12 |
| Total | 120 | 80 |
| B1 | for A values | |
| B1 | for B values |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): no association between area and age profile | B1 | At least \(H_0\) |
| \(H_1\): association between area and age profile | ||
| \(O_i\) | \(E_i\) | \(\frac{(O_i - E_i)^2}{E_i}\) |
| 24 | 31.8 | 3.6679 |
| 72 | 64.8 | 0.8000 |
| 24 | 23.4 | 0.5538 |
| 32 | 21.2 | 5.5019 |
| 36 | 43.2 | 1.2000 |
| 12 | 15.6 | 0.8308 |
| M1 | Attempt at Row & Column totals | |
| M1 | Attempt at \(E_i\) | |
| M1 | Attempt at \(\frac{(O_i - E_i)^2}{E_i}\) | |
| M1 | Attempt at \(\chi^2\) | |
| \(\sum O_i = 200\), \(\sum E_i = 200\), \(\chi^2 = 12.554\) | A1 | AWFW 12.5 to 12.6 provided correct method used |
| \(v = (3-1)(2-1) = 2\) | B1 | |
| \(\chi^2_{1\%}(2) = 9.210 < 12.554\) | B1 | ft on their \(v\) and \(\chi^2\) |
| Reject \(H_0\) | ||
| The evidence suggests that the area within which a school is situated seems to have an effect on the age-profile of the staff employed. | E1 | ft on \(\chi^2\) and calculated value |
| Answer | Marks |
|---|---|
| There seems to be fewer staff employed in 22-34 age group than expected in school A and more than expected in school B | E1, E1 |
## 4(a)(i)
| | A | B | Total |
|---|---|---|---|
| 22-34 | 21 | 32 | 53 |
| 35-39 | 72 | 36 | 108 |
| 40-59 | 27 | 12 | 39 |
| Total | 120 | 80 | 200 |
| B1 | for A values
| B1 | for B values
## 4(a)(ii)
$H_0$: no association between area and age profile | B1 | At least $H_0$
$H_1$: association between area and age profile |
| $O_i$ | $E_i$ | $\frac{(O_i - E_i)^2}{E_i}$ |
|---|---|---|
| 24 | 31.8 | 3.6679 |
| 72 | 64.8 | 0.8000 |
| 24 | 23.4 | 0.5538 |
| 32 | 21.2 | 5.5019 |
| 36 | 43.2 | 1.2000 |
| 12 | 15.6 | 0.8308 |
| M1 | Attempt at Row & Column totals
| M1 | Attempt at $E_i$
| M1 | Attempt at $\frac{(O_i - E_i)^2}{E_i}$
| M1 | Attempt at $\chi^2$
$\sum O_i = 200$, $\sum E_i = 200$, $\chi^2 = 12.554$ | A1 | AWFW 12.5 to 12.6 provided correct method used
$v = (3-1)(2-1) = 2$ | B1 |
$\chi^2_{1\%}(2) = 9.210 < 12.554$ | B1 | ft on their $v$ and $\chi^2$
Reject $H_0$ |
The evidence suggests that the area within which a school is situated seems to have an effect on the age-profile of the staff employed. | E1 | ft on $\chi^2$ and calculated value
## 4(b)
There seems to be fewer staff employed in 22-34 age group than expected in school A and more than expected in school B | E1, E1 |
---4 It is claimed that the area within which a school is situated affects the age profile of the staff employed at that school. In order to investigate this claim, the age profiles of staff employed at two schools with similar academic achievements are compared.
Academia High School, situated in a rural community, employs 120 staff whilst Best Manor Grammar School, situated in an inner-city community, employs 80 staff.
The percentage of staff within each age group, for each school, is given in the table.
\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
Age & \begin{tabular}{ c }
Academia \\
High School \\
\end{tabular} & \begin{tabular}{ c }
Best Manor \\
Grammar School \\
\end{tabular} \\
\hline
$\mathbf { 2 2 - } \mathbf { 3 4 }$ & 17.5 & 40.0 \\
\hline
$\mathbf { 3 5 - } \mathbf { 3 9 }$ & 60.0 & 45.0 \\
\hline
$\mathbf { 4 0 - } \mathbf { 5 9 }$ & 22.5 & 15.0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Form the data into a contingency table suitable for analysis using a $\chi ^ { 2 }$ distribution.\\
(2 marks)
\item Use a $\chi ^ { 2 }$ test, at the $1 \%$ level of significance, to determine whether there is an association between the age profile of the staff employed and the area within which the school is situated.
\end{enumerate}\item Interpret your result in part (a)(ii) as it relates to the 22-34 age group.
\end{enumerate}
\hfill \mbox{\textit{AQA S2 2006 Q4 [13]}}