The number of successes in n trials is to be used to test the null hypothesis that the parameter θ of a bino-mial population equals 1 2 against the alternative that it does not equal 12 . (a) Find an expression for the likelihood ratio statistic.(b) Use the result of part (a) to show that the criticalregion of the likelihood ratio test can be written asx · ln x + (n − x)· ln(n − x) G Kwhere x is the observed number of successes.(c) Study the graph of f(x) = x · ln x + (n − x)· ln(n − x),in particular its minimum and its symmetry, to show thatthe critical region of this likelihood ratio test can also bewritten as x − n2G K where K is a constant that depends on the size of the crit-ical region.
The number of successes in n trials is to be used to
test the null hypothesis that the parameter θ of a bino-
mial 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
x · ln x + (n − x)· ln(n − x) G K
where x is the observed number of successes.
(c) Study the graph of f(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
x − n
2
G K
where K is a constant that depends on the size of the crit-
ical region.
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