Suppose three basketball players shoot n1, n2, and n3 free throws and make Y1, Y2, and Y3 of them, respectively. Regard the free throws of each player as independent Bernoulli trials and denote the probability of successes of the three players by p1, p2, and p3. You are interested in whether the players are unequally skilled at free throws.
(a) Give null and alternate hypotheses of interest.
(b) Give the likelihood function L(p1, p2, p3; Y1, Y2, Y3) for p1, p2, and p3.
(c) Give the log-likelihood function l(p1, p2, p3; Y1, Y2, Y3).
(d) Give an expression for pˆ0 = argmax p?[0,1] `(p, p, p; Y1, Y2, Y3).
(e) Give an expression for the likelihood ratio.
(f) Give an expression for the test statistic of the asymptotic LRT for the hypotheses in part (a).
(g) Give the degrees of freedom of the chi-squared distribution relevant to the asymptotic LRT.
(h) Suppose n1 = n2 = n3 = 100 and Y1 = 75, Y2 = 80, and Y3 = 72.
i. Compute the test statistic for the asymptotic likelihood ratio test.
ii. Give the p-value for the asymptotic likelihood ratio test.
iii. Interpret your results. Would you say the three players are unequally skilled.
Sample Solution