| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Calculating bias of estimator |
| Difficulty | Standard +0.3 This is a straightforward S4 question on bias and variance of estimators requiring routine application of E(θ) and Var(θ) formulas. Students need to recognize unbiased estimators (E(θ) = μ) and calculate variances using independence, then compare using standard criteria (unbiased + minimum variance). All steps are mechanical with no novel insight required, making it slightly easier than average. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
A random sample $X_1, X_2, ..., X_{10}$ is taken from a population with mean $\mu$ and variance $\sigma^2$.
\begin{enumerate}[label=(\alph*)]
\item Determine the bias, if any, of each of the following estimators of $\mu$.
$$\theta_1 = \frac{X_1 + X_4 + X_5}{3}$$
$$\theta_2 = \frac{X_{10} - X_1}{3}$$
$$\theta_3 = \frac{3X_1 + 2X_5 + X_{10}}{6}$$
[4]
\item Find the variance of each of these estimators.
[5]
\item State, giving reasons, which of these three estimators for $\mu$ is
\begin{enumerate}[label=(\roman*)]
\item the best estimator,
\item the worst estimator.
\end{enumerate}
[4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 Q1 [13]}}