A random sample of two observations \(X _ { 1 }\) and \(X _ { 2 }\) is taken from a population with unknown mean \(\mu\) and unknown variance \(\sigma ^ { 2 }\)
Explain why \(\frac { X _ { 1 } - \mu } { \sigma }\) is not a statistic.
Explain what you understand by an unbiased estimator for \(\mu\)
Two estimators for \(\mu\) are \(U _ { 1 }\) and \(U _ { 2 }\) where
$$U _ { 1 } = 3 X _ { 1 } - 2 X _ { 2 } \quad \text { and } \quad U _ { 2 } = \frac { X _ { 1 } + 3 X _ { 2 } } { 4 }$$
Show that both \(U _ { 1 }\) and \(U _ { 2 }\) are unbiased estimators for \(\mu\)
The most efficient estimator among a group of unbiased estimators is the one with the smallest variance.
By finding the variance of \(U _ { 1 }\) and the variance of \(U _ { 2 }\) state, giving a reason, the most efficient estimator for \(\mu\) from these two estimators.