Question

Two-hundred years ago it was established that the proportion of female births in European populations was less than 0.5. You are interested in learning about the proportion of female births in Paris between 1145 and 1770. Data collected during that time show that 241,945 girls and 251,527 boys were born in Paris during that time period. Assume the number of female births follows a binomial distribution.

(a) Compute L(0.5), or the “likelihood” that the probability of a female birth is 0.5.

(b) Plot the likelihood function of p for p ∈ [0.48, 0.51].

(c) What is the value of the maximum likelihood estimate of p?

Answer 1

(a)

Total births = 241,945 + 251,527 = 493472

Assume the number of female births follows a binomial distribution,  number of female births F ~ (n = 493472, p)

Given there were 241,945 girls among the total births.

Using binomial distribution, the likelihood function is,

L(p) =

For p = 0.5

= 4.474383 x 10^-44 (Using calculator)

(b)

The plot of likelihood function of p for p ∈ [0.48, 0.51] is,

(c)

We see that the likelihood function is maximum for p = 0.49. Thus, the value of the maximum likelihood estimate of p is 0.49