| Exam Board | WJEC |
|---|---|
| Module | Further Unit 2 (Further Unit 2) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Expected frequencies partially provided |
| Difficulty | Moderate -0.3 This is a standard chi-squared test question requiring routine calculations: finding expected values using row/column totals, and computing chi-squared contributions using the formula (O-E)²/E. While it involves multiple steps and careful arithmetic, it requires no novel insight—just direct application of well-practiced formulas from the statistics syllabus. |
| Spec | 5.06a Chi-squared: contingency tables |
| Fewer than 5 GCSE | 5 or more GCSE | 3 A Levels | University degree | Post graduate qualification | Total | |
| Less than £20 000 | 18 | 32 | 20 | 28 | 10 | 108 |
| £20 000 to £60 000 | 50 | 95 | 112 | 155 | 50 | 462 |
| More than £60 000 | 3 | 22 | 29 | 35 | 5 | 94 |
| Total | 71 | 149 | 161 | 218 | 65 | 664 |
| Expected values | Fewer than 5 GCSE | 5 or more GCSE | 3 A Levels | University degree | Post graduate qualification |
| Less than £20 000 | \(k\) | 24·23 | 26·19 | 35·46 | 10·57 |
| £20 000 to £60 000 | 49·40 | 103·67 | 112·02 | 151·68 | 45·23 |
| More than £60 000 | 10·05 | 21·09 | 22·79 | 30·86 | 9·20 |
| Chi Squared Contributions | Fewer than 5 GCSE | 5 or more GCSE | 3 A Levels | University degree | Post graduate qualification |
| Less than £20 000 | 3·604530799 | \(m\) | 1·46165 | 1·5686 | 0·03098 |
| £20 000 to £60 000 | 0·007272735 | 0·72535 | 4E-06 | 0·07264 | 0·50396 |
| More than £60 000 | 4·946619863 | 0·03897 | 1·69081 | 0·55498 | \(n\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0\): There is no association between highest level of education and salary. | B1 | OR \(H_0\): Highest level of education and salary are independent. \(H_1\): Highest level of education and salary are not independent. |
| Answer | Marks | Guidance |
|---|---|---|
| \(k = \frac{108 \times 71}{664} = 11.54(8 \ldots)\) or 11.55 | M1 A1 | Alternative method: 71 – (49.4 + 10.05) OR 108 – (10.57 + 35.46 + 26.19 + 24.23) |
| Answer | Marks | Guidance |
|---|---|---|
| \(m = \frac{(32 - 24.23)^2}{24.23}\) \(m = 2.49166\) Accept 2.491659.... | M1 | M1 either method correct. |
| \(n = \frac{(5 - 9.20)^2}{9.20}\) \(n = 1.91739\) Accept 1.917391.... | A1 | Both correct. NB Using more dp than in the expected values table gives 2.48798 and 1.91866 |
| Answer | Marks |
|---|---|
| Add the chi squared contributions to get 19.61301 | E1 |
| Answer | Marks |
|---|---|
| Appropriate comment relating observed and expected values. Eg. Fewer than expected in the highest earning category. More than expected in the lowest earning category. Expected value does not deviate much from observed value for £20 000 - £60 000 but does for the other two. | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| Appropriate comment on \(p\) value. Eg. The \(p\) value is < 0.05 which implies there is an association between highest level of education attained and salary. | E1 | May be given for 19.61 > 15.507 implies there is an association between highest level of education attained and salary. |
| e.g. At the 1% significance level there is no association between highest level of education attained and salary. | ||
| e.g. Although it can be shown there is an association it does not imply that highest level of education attained leads to a higher paying job. | [8] |
## 6(a)
$H_0$: There is no association between highest level of education and salary. | B1 | OR $H_0$: Highest level of education and salary are independent. $H_1$: Highest level of education and salary are not independent.
$H_1$: There is an association between highest level of education and salary.
## 6(b)
$k = \frac{108 \times 71}{664} = 11.54(8 \ldots)$ or 11.55 | M1 A1 | Alternative method: 71 – (49.4 + 10.05) OR 108 – (10.57 + 35.46 + 26.19 + 24.23)
## 6(c)
$m = \frac{(32 - 24.23)^2}{24.23}$ $m = 2.49166$ Accept 2.491659.... | M1 | M1 either method correct.
$n = \frac{(5 - 9.20)^2}{9.20}$ $n = 1.91739$ Accept 1.917391.... | A1 | Both correct. NB Using more dp than in the expected values table gives 2.48798 and 1.91866
## 6(d)(i)
Add the chi squared contributions to get 19.61301 | E1 |
## 6(d)(ii)
Appropriate comment relating observed and expected values. Eg. Fewer than expected in the highest earning category. More than expected in the lowest earning category. Expected value does not deviate much from observed value for £20 000 - £60 000 but does for the other two. | E1 |
## 6(e)
Appropriate comment on $p$ value. Eg. The $p$ value is < 0.05 which implies there is an association between highest level of education attained and salary. | E1 | May be given for 19.61 > 15.507 implies there is an association between highest level of education attained and salary.
e.g. At the 1% significance level there is no association between highest level of education attained and salary. | | |
e.g. Although it can be shown there is an association it does not imply that highest level of education attained leads to a higher paying job. | [8] |
---A student, considering options for the future, collects data on education and salary. The table below shows the highest level of education attained and the salary bracket of a random sample of 664 people.
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
& Fewer than 5 GCSE & 5 or more GCSE & 3 A Levels & University degree & Post graduate qualification & Total \\
\hline
Less than £20 000 & 18 & 32 & 20 & 28 & 10 & 108 \\
\hline
£20 000 to £60 000 & 50 & 95 & 112 & 155 & 50 & 462 \\
\hline
More than £60 000 & 3 & 22 & 29 & 35 & 5 & 94 \\
\hline
Total & 71 & 149 & 161 & 218 & 65 & 664 \\
\hline
\end{tabular}
By conducting a chi-squared test for independence, the student investigates the relationship between the highest level of education attained and the salary earned.
\begin{enumerate}[label=(\alph*)]
\item State the null and alternative hypotheses. [1]
\item The table below shows the expected values. Calculate the value of $k$. [2]
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Expected values & Fewer than 5 GCSE & 5 or more GCSE & 3 A Levels & University degree & Post graduate qualification \\
\hline
Less than £20 000 & $k$ & 24·23 & 26·19 & 35·46 & 10·57 \\
\hline
£20 000 to £60 000 & 49·40 & 103·67 & 112·02 & 151·68 & 45·23 \\
\hline
More than £60 000 & 10·05 & 21·09 & 22·79 & 30·86 & 9·20 \\
\hline
\end{tabular}
\item The following computer output is obtained. Calculate the values of $m$ and $n$. [2]
\begin{tabular}{|l|c|c|c|c|c|}
\hline
Chi Squared Contributions & Fewer than 5 GCSE & 5 or more GCSE & 3 A Levels & University degree & Post graduate qualification \\
\hline
Less than £20 000 & 3·604530799 & $m$ & 1·46165 & 1·5686 & 0·03098 \\
\hline
£20 000 to £60 000 & 0·007272735 & 0·72535 & 4E-06 & 0·07264 & 0·50396 \\
\hline
More than £60 000 & 4·946619863 & 0·03897 & 1·69081 & 0·55498 & $n$ \\
\hline
\end{tabular}
X-squared = 19·61301, df = 8, p-value = 0·0119
\item \begin{enumerate}[label=(\roman*)]
\item Without carrying out any further calculations, explain how X-squared = 19·61301 (the $\chi^2$ test statistic) was calculated. [2]
\item Comment on the values in the "Fewer than 5 GCSE" column of the table in part (c). [2]
\end{enumerate}
\item The student says that the highest levels of education lead to the highest paying jobs. Comment on the accuracy of the student's statement. [1]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 2 2018 Q6 [10]}}