| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Year | 2024 |
| Session | June |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Variance of estimators |
| Difficulty | Challenging +1.8 This is a substantial Further Maths statistics question requiring multiple techniques: deriving moments from a pdf, proving unbiasedness, computing variance of estimators, finding standard errors, and comparing estimator efficiency. Part (a)(ii) requires number-theoretic insight about when √n divides α² to yield rational standard errors. The multi-part structure with 19 total marks and the need to work with abstract parameter α elevates this above routine A-level questions, though the individual techniques are standard for Further Maths S2/S3 content. |
| Spec | 5.03b Solve problems: using pdf5.05b Unbiased estimates: of population mean and variance |
The probability density function of the continuous random variable $X$ is given by
$$f(x) = \frac{3x^2}{\alpha^3} \quad \text{for } 0 \leq x \leq \alpha$$
$$f(x) = 0 \quad \text{otherwise.}$$
$\overline{X}$ is the mean of a random sample of $n$ observations of $X$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that $U = \frac{4\overline{X}}{3}$ is an unbiased estimator for $\alpha$. [5]
\item If $\alpha$ is an integer, what is the smallest value of $n$ that gives a rational value for the standard error of $U$? [9]
\end{enumerate}
\item $\overline{X}_1$ and $\overline{X}_2$ are the means of independent random samples of $X$, each of size $n$.
The estimator $V = 4\overline{X}_1 - \frac{8}{3}\overline{X}_2$ is also an unbiased estimator for $\alpha$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{\text{Var}(U)}{\text{Var}(V)} = \frac{1}{13}$. [4]
\item Hence state, with a reason, which of $U$ or $V$ is the better estimator. [1]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 5 2024 Q5 [19]}}