| Exam Board | WJEC |
|---|---|
| Module | Further Unit 2 (Further Unit 2) |
| Year | 2024 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Cell combining required |
| Difficulty | Moderate -0.5 This is a standard chi-squared test of independence with straightforward calculations. Students must complete expected frequency tables using row×column/total, calculate chi-squared contributions using (O-E)²/E, and compare to critical values. While it involves multiple calculations and a 5×3 contingency table (larger than typical), the methodology is entirely routine for Further Maths statistics with no conceptual challenges or novel problem-solving required. |
| Spec | 5.06a Chi-squared: contingency tables |
| Learned Welsh | ||||
| Attitude | From two parents/carers | From one parent/carer | At school only | Total |
| Very negative | 2 | 14 | 30 | 46 |
| Slightly negative | 4 | 20 | 21 | 45 |
| Neutral | 12 | 17 | 8 | 37 |
| Slightly positive | 21 | 19 | 11 | 51 |
| Very positive | 25 | 21 | 28 | 74 |
| Total | 64 | 91 | 98 | 253 |
| Expected Frequencies | Learned Welsh | ||
| Attitude | From two parents/carers | From one parent/carer | At school only |
| Very negative | 11.64 | 16.55 | 17.82 |
| Slightly negative | 11.38 | 16.19 | 17.43 |
| Neutral | 9.36 | 13.31 | 14.33 |
| Slightly positive | 12.90 | 18.34 | 19.75 |
| Very positive | F | 26.62 | 28.66 |
| Chi-Squared Contributions | Learned Welsh | ||
| Attitude | From two parents/carers | From one parent/carer | At school only |
| Very negative | 7.98 | 0.39 | 8.33 |
| Slightly negative | \(4 \cdot 79\) | 0.90 | 0.73 |
| Neutral | \(0 \cdot 74\) | 1.02 | G |
| Slightly positive | 5.08 | 0.02 | 3.88 |
| Very positive | 2.11 | 1.19 | 0.02 |
| Total | 20.70 | 3.52 | H |
5. Lily is interested in the relationship between the way in which students learned Welsh and their attitude towards the Welsh language.
Students were categorised as having learned Welsh in one of three ways:
\begin{itemize}
\item from one Welsh-speaking parent/carer at home,
\item from two Welsh-speaking parents/carers at home,
\item at school only, for those with no Welsh-speaking parents/carers at home.
\end{itemize}
The students were asked to rate their attitude towards the Welsh language from 'Very negative' to 'Very positive'.
The following data for a random sample of 253 students were collected as part of a project.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
& \multicolumn{3}{|c|}{Learned Welsh} & \\
\hline
Attitude & From two parents/carers & From one parent/carer & At school only & Total \\
\hline
Very negative & 2 & 14 & 30 & 46 \\
\hline
Slightly negative & 4 & 20 & 21 & 45 \\
\hline
Neutral & 12 & 17 & 8 & 37 \\
\hline
Slightly positive & 21 & 19 & 11 & 51 \\
\hline
Very positive & 25 & 21 & 28 & 74 \\
\hline
Total & 64 & 91 & 98 & 253 \\
\hline
\end{tabular}
\end{center}
Lily intends to carry out a chi-squared test for independence at the $5 \%$ level. She produces the following tables which are incomplete.
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Expected Frequencies & \multicolumn{3}{|c|}{Learned Welsh} \\
\hline
Attitude & From two parents/carers & From one parent/carer & At school only \\
\hline
Very negative & 11.64 & 16.55 & 17.82 \\
\hline
Slightly negative & 11.38 & 16.19 & 17.43 \\
\hline
Neutral & 9.36 & 13.31 & 14.33 \\
\hline
Slightly positive & 12.90 & 18.34 & 19.75 \\
\hline
Very positive & F & 26.62 & 28.66 \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Chi-Squared Contributions & \multicolumn{3}{|c|}{Learned Welsh} \\
\hline
Attitude & From two parents/carers & From one parent/carer & At school only \\
\hline
Very negative & 7.98 & 0.39 & 8.33 \\
\hline
Slightly negative & $4 \cdot 79$ & 0.90 & 0.73 \\
\hline
Neutral & $0 \cdot 74$ & 1.02 & G \\
\hline
Slightly positive & 5.08 & 0.02 & 3.88 \\
\hline
Very positive & 2.11 & 1.19 & 0.02 \\
\hline
Total & 20.70 & 3.52 & H \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Calculate the values of $F , G$ and $H$.
\item Carry out Lily's chi-squared test for independence at the $5 \%$ level.
\item By referring to the figures in the tables on pages 16 and 17, give two comments on the relationship between the way students learned Welsh and their attitude towards the Welsh language.
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 2 2024 Q5 [12]}}