1. Two organisations are each asked to carry out a survey to find out the proportion, \(p\), of the population that would vote for a particular political party.

The first organisation finds that out of \(m\) people, \(X\) would vote for this particular political party. The second organisation finds that out of \(n\) people, \(Y\) would vote for this particular political party. An unbiased estimator, \(Q\), of \(p\) is proposed where $$Q = k \left( \frac { X } { m } + \frac { Y } { n } \right)$$

1. Show that \(k = \frac { 1 } { 2 }\) A second unbiased estimator, \(R\), of \(p\) is proposed where $$R = \frac { a X } { m } + \frac { b Y } { n }$$
2. Show that \(a + b = 1\) Given that \(m = 100\) and \(n = 200\) and that \(R\) is a better estimator of \(p\) than \(Q\)
3. calculate the range of possible values of \(a\) Show your working clearly.