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Question

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&gt;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.

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