Question: 6. The data in the table below are the number of females and males in the progeny of 16 female light brown apple moths in Muswellbrook, New South Wales, Australia.

(a) Calculate the proportion of females in each of the 16 groups of progeny

(b) Let \({{Y}_{i}}\) denote the number of females and \({{n}_{i}}\) the number of progeny in each group. Suppose the \({{Y}_{i}}\) ’s are independent random variables each with Binomial Distribution

\[f\left( {{y}_{i}};\theta \right)=\left( \begin{matrix} {{n}_{i}} \\ {{y}_{i}} \\ \end{matrix} \right){{\theta }^{{{y}_{i}}}}{{\left( 1-\theta \right)}^{{{n}_{i}}-{{y}_{i}}}}\]

Find the MLE of \(\theta \) using calculus and evaluating the data.