Imagine that you wish to estimate the fraction of a population that is female. Suppose you take a random sample ofnindividuals from the population, andkof them are female. Suppose further that the population is large enough relative tonthat your sample has a negligible effect on the fraction of females in the remaining population. In this case we can model the number of females in the sample as a binomial random variable with unknownparameterp.(a) What is the probability of obtaining the observed sample as a function ofp?

P(X=k) =nCk *p^k (1p)^nk

The function obtained in part (a) is referred to as alikelihood function. Themaximum likelihood estimateforpis the value ofpthat maximizes the likelihood function. What is the maximum likelihood estimate of the fraction of females in thepopulationp,where the 'hat' indicates that this value ofpis the maximum likelihood estimate?