3. A population has mean \(\mu\) and variance \(\sigma ^ { 2 }\). A random sample of size 3 is to be taken from this population and \(\bar { X }\) denotes its sample mean. A second random sample of size 4 is to be taken from this population and \(\bar { Y }\) denotes its sample mean.

1. Show that unbiased estimators for \(\mu\) are given by
   1. \(\hat { \mu } _ { 1 } = \frac { 1 } { 3 } \bar { X } + \frac { 2 } { 3 } \bar { Y }\),
   2. \(\hat { \mu } _ { 2 } = \frac { 5 \bar { X } + 4 \bar { Y } } { 9 }\).
2. Calculate Var \(\left( \hat { \mu } _ { 1 } \right)\)
3. Given that \(\operatorname { Var } \left( \hat { \mu } _ { 2 } \right) = \frac { 37 } { 243 } \sigma ^ { 2 }\), state, giving a reason, which of these two estimators should be
   used. used.