| Exam Board | WJEC |
|---|---|
| Module | Unit 4 (Unit 4) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Direct expected frequency calculation |
| Difficulty | Moderate -0.8 This is a straightforward statistics question requiring basic normal distribution calculations (finding probabilities and expected frequencies), followed by simple commentary on model fit. The calculations are routine A-level statistics content with no conceptual challenges, and the interpretation parts require only superficial observations about the given data rather than deep statistical reasoning. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Cost, \(x\) (£) | \(x < 40\) | \(40 \leqslant x<50\) | \(50 \leqslant x<60\) | \(60 \leqslant x<70\) | \(70 \leqslant x<80\) | \(80 \leqslant x<90\) | \(x \geqslant 90\) |
| Number of households | 5 | 11 | 16 | 18 | 15 | 9 | 6 |
Arwyn collects data about household expenditure on food. He records the weekly expenditure on food for 80 randomly selected households from across Wales.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Cost, $x$ (£) & $x < 40$ & $40 \leqslant x<50$ & $50 \leqslant x<60$ & $60 \leqslant x<70$ & $70 \leqslant x<80$ & $80 \leqslant x<90$ & $x \geqslant 90$ \\
\hline
Number of households & 5 & 11 & 16 & 18 & 15 & 9 & 6 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Explain why a normal distribution may be an appropriate model for the weekly expenditure on food for this sample. [1]
\end{enumerate}
Arwyn uses the distribution N(64, 15²) to model expenditure on food.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the number of households in the sample that this model would predict to have weekly food expenditure in the range
\begin{enumerate}[label=(\roman*)]
\item $60 \leqslant x < 70$,
\item $x \geqslant 90$. [4]
\end{enumerate}
\item Use your answers to part (b)
\begin{enumerate}[label=(\roman*)]
\item to comment on the suitability of this model,
\item to explain how Arwyn could improve the model by changing one of its parameters. [2]
\end{enumerate}
\item Arwyn's friend Colleen wishes to use the improved model to predict household expenditure on food in Northern Ireland. Comment on this plan. [1]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 4 2018 Q4 [8]}}