| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Session | Specimen |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Direct comparison with scalar multiple (different variables) |
| Difficulty | Standard +0.8 This is a Further Maths statistics question requiring multiple applications of normal distribution theory including quartile calculation, sum of independent normals, and linear combinations of normals (X - 2Y). While the individual techniques are standard, part (b) requires recognizing that X - 2Y follows a normal distribution and correctly computing its parameters, which demands solid conceptual understanding beyond routine application. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.04b Linear combinations: of normal distributions |
Alun does the crossword in the Daily Bugle every day. The time that he takes to complete the crossword, $X$ minutes, is modelled by the normal distribution $\mathrm{N}(32, 4^2)$. You may assume that the times taken to complete the crossword on successive days are independent.
\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Find the upper quartile of $X$ and explain its meaning in context.
\item Find the probability that the total time taken by Alun to complete the crosswords on five randomly chosen days is greater than 170 minutes. [7]
\end{enumerate}
\item Belle also does the crossword every day and the time that she takes to complete the crossword, $Y$ minutes, is modelled by the normal distribution $\mathrm{N}(18, 2^2)$. Find the probability that, on a randomly chosen day, the time taken by Alun to complete the crossword is more than twice the time taken by Belle to complete the crossword. [6]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 5 Q1 [13]}}