Question 1 - A-Level Maths

Exam Board	Edexcel
Module	S4 (Statistics 4)
Year	2008
Session	June
Topic	Central limit theorem
Type	Estimator properties and bias

A random sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { 10 }$ is taken from a population with mean $\mu$ and variance $\sigma ^ { 2 }$.
1. Determine the bias, if any, of each of the following estimators of $\mu$.
$$\begin{aligned} & \theta _ { 1 } = \frac { X _ { 3 } + X _ { 4 } + X _ { 5 } } { 3 }
& \theta _ { 2 } = \frac { X _ { 10 } - X _ { 1 } } { 3 }
& \theta _ { 3 } = \frac { 3 X _ { 1 } + 2 X _ { 2 } + X _ { 10 } } { 6 } \end{aligned}$$
Find the variance of each of these estimators.
State, giving reasons, which of these three estimators for $\mu$ is
1. the best estimator,
2. the worst estimator.