The number of successes in n trials is to be used to test the null hypothesis that the parameter θ of a binomial population equals 1/2 against the alternative that it does not equal 1/2.
(a) Find an expression for the likelihood ratio statistic.
(b) Use the result of part (a) to show that the critical region of the likelihood ratio test can be written as
where x is the observed number of successes.
(c) Study the graph of 
(x) = x · ln x + (n - x)· ln(n - x), in particular its minimum and its symmetry, to show that the critical region of this likelihood ratio test can also be written as
where K is a constant that depends on the size of the critical region.