Suppose X ∼ Binomial(100, p), and we make the observation X = x. Say whether each of the following statements is true or false. If the statement is false, explain why.

a. The likelihood function is L(p ; x) =

100 x

p x

(1 − p)

100−x for 0 ≤ p ≤ 1.

b. The probability function is fX(x) =

100 x

p x

(1 − p)

100−x for x = 0, 1, . . . , 100.

c. If a test of H0 : p = 0.5 is not significant, the maximum likelihood estimate of p must be pb= 0.5.

d. The maximum likelihood estimate is x/100 and the maximum likelihood estimator is X/10