Before an election, combining the results of 12,625 polls with 14,491,635

samples in total, it shows that 6,413,959 responders (44.3%) say they will

vote for the ﬁrst candidate and 6,134,272 responders (42.3%) say they will

vote for the other candidate. Assume a binomial model Binomial(n, p) of

the polls for the ﬁrst and second candidates, where p is the percentage of

the votes to the ﬁrst candidate and n is the total numb er of votes to the ﬁrst

candidate or the second candidate. Suppose we are interested in whether

the ﬁrst candidate wins more than half of the votes to the ﬁrst and second candidates:

H0: p=0.5v.s. H1: p>0.5 (a) Compute the test statistics of the generalized likelihood ratio test. Is

this test a uniformly most powerful test? (b) Use Wilks’ theorem to compute the critical value of the generalized

likelihood ratio test under or = 0.05 level. Make a decision. .a. (c) Another test has test statistics %, where p0 = 0.5. Compute

P0 ‘P0 “- the p-value of this test using the central limit theorem and make a

decision. Assume the signiﬁcance level or = 0.05. (d) If the second candidate wins the election, comment on possible prob—

lems in this statistical analysis.