7 The continuous random variable \(U\) has unknown mean \(\mu\) and known variance \(\sigma ^ { 2 }\). In order to estimate \(\mu\), two random samples, one of 4 observations of \(U\) and the other of 6 observations of \(U\), are taken. The sample means are denoted by \(\bar { U } _ { 4 }\) and \(\bar { U } _ { 6 }\) respectively. One estimator \(S\), given by \(S = \frac { 1 } { 2 } \left( \bar { U } _ { 4 } + \bar { U } _ { 6 } \right)\), is proposed.

1. Show that \(S\) is unbiased and find \(\operatorname { Var } ( S )\) in terms of \(\sigma ^ { 2 }\). A second estimator \(T\) of the form \(a \bar { U } _ { 4 } + b \bar { U } _ { 6 }\) is proposed, where \(a\) and \(b\) are chosen such that \(T\) is an unbiased estimator for \(\mu\) with the smallest possible variance.
2. Find the values of \(a\) and \(b\) and the corresponding variance of \(T\).
3. State, giving a reason, which of \(S\) and \(T\) is the better estimator.
4. Compare the efficiencies of this preferred estimator and the mean of all 10 observations.