Answered: The number of successes in n trials is…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

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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