1.6 The data in Table 1.4 are the numbers of females and males in the progeny of 16 female light brown apple moths in Muswellbrook, New South Wales, Australia (from Lewis, 1987).

a. Calculate the proportion of females in each of the 16 groups of progeny.

b. Let Yi denote the number of females and ni the number of progeny in each group (i = 1,...,16). Suppose the Yi’s are independent random variables each with the Binomial distribution f(yi

;θ) = 

ni yi



θ

yi

(1−θ)

ni−yi

.

Find the maximum likelihood estimator of θ using calculus and evaluate it for these data.

c. Use a numerical method to estimate θb and compare the answer with the one from (b)