Standard +0.8 This question requires understanding of unbiased estimators and variance calculations for linear combinations of sample means. Part (i) is definitional recall. Part (ii)(a) requires using E(T₁)=μ to find a=4. Part (ii)(b) involves standard variance calculations with independence. Part (ii)(c) requires comparing expressions. While systematic, it demands careful algebraic manipulation across multiple parts and understanding of statistical properties—moderately challenging for A-level standard but accessible with solid technique.
The statistic \(T\) is derived from a random sample taken from a population which has an unknown parameter \(\theta\). \(T\) is an unbiased estimator of \(\theta\). What does the statement ' \(T\) is an unbiased estimator of \(\theta ^ { \prime }\) imply?
A random sample of size \(n\) is taken from each of two independent populations. The first population has a non-zero mean \(\mu\) and variance \(\sigma ^ { 2 }\) and \(\bar { X } _ { 1 }\) denotes the sample mean. The second population has mean \(\frac { 1 } { 2 } \mu\) and variance \(b \sigma ^ { 2 }\), where \(b\) is a positive constant, and \(\bar { X } _ { 2 }\) denotes the sample mean. Two unbiased estimators for \(\mu\) are defined by
$$T _ { 1 } = 3 \bar { X } _ { 1 } - a \bar { X } _ { 2 } \quad \text { and } \quad T _ { 2 } = \frac { 1 } { 5 } \left( 4 \bar { X } _ { 1 } + 2 \bar { X } _ { 2 } \right) .$$
Determine the value of \(a\).
Show that \(\operatorname { Var } \left( T _ { 1 } \right) = \frac { \sigma ^ { 2 } } { n } ( 9 + 16 b )\) and find a similar expression for \(\operatorname { Var } \left( T _ { 2 } \right)\).
The estimator with the smaller variance is preferred. State which of \(T _ { 1 }\) and \(T _ { 2 }\) is the preferred estimator of \(\mu\).
2 (i) The statistic $T$ is derived from a random sample taken from a population which has an unknown parameter $\theta$. $T$ is an unbiased estimator of $\theta$. What does the statement ' $T$ is an unbiased estimator of $\theta ^ { \prime }$ imply?\\
(ii) A random sample of size $n$ is taken from each of two independent populations. The first population has a non-zero mean $\mu$ and variance $\sigma ^ { 2 }$ and $\bar { X } _ { 1 }$ denotes the sample mean. The second population has mean $\frac { 1 } { 2 } \mu$ and variance $b \sigma ^ { 2 }$, where $b$ is a positive constant, and $\bar { X } _ { 2 }$ denotes the sample mean. Two unbiased estimators for $\mu$ are defined by
$$T _ { 1 } = 3 \bar { X } _ { 1 } - a \bar { X } _ { 2 } \quad \text { and } \quad T _ { 2 } = \frac { 1 } { 5 } \left( 4 \bar { X } _ { 1 } + 2 \bar { X } _ { 2 } \right) .$$
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $a$.
\item Show that $\operatorname { Var } \left( T _ { 1 } \right) = \frac { \sigma ^ { 2 } } { n } ( 9 + 16 b )$ and find a similar expression for $\operatorname { Var } \left( T _ { 2 } \right)$.
\item The estimator with the smaller variance is preferred. State which of $T _ { 1 }$ and $T _ { 2 }$ is the preferred estimator of $\mu$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2013 Q2}}