Applied Statistics and Probability for Engineers

9-3

Notice that is the square of the value of a random variable that has the

t-distribution with n 1 degrees of freedom when the null hypothesis H 0 :  0 is true. So

we may write the value of the likelihood ratio as

It is easy to find the value for the constant kthat would lead to rejection of the null hypothe-

sisH 0. Since we reject H 0 if  k, this implies that small values of support the alternative

hypothesis. Clearly, will be small when t^2 is large. So instead of specifying kwe can spec-

ify a constant cand reject H 0 :  0 if t^2 c. The critical values of twould be the extreme

values, either positive or negative, and if we wish to control the type I error probability at ,

the critical region in terms of twould be

or, equivalently, we would reject H 0 :  0 if t^2 c. Therefore, the likelihood

ratio test for H 0 :  0 versus H 1 :  0 is the familiar single-sample t-test.

The procedure employed in this example to find the critical region for the likelihood ratio

is used often. That is, typically, we can manipulate to produce a condition that is equiva-

lent to     k, but one that is simpler to use.

The likelihood ratio principle is a very general procedure. Most of the tests presented in

this book that utilize the t, chi-square, and F-distributions for testing means and variances of

normal distributions are likelihood ratio tests. The principle can also be used in cases where

the observations are dependent, or even in cases where their distributions are different.

However, the likelihood function can be very complicated in some of these situations. To use

the likelihood principle we must specify the form of the distribution. Without such a specifi-

cation, it is impossible to write the likelihood function, and so if we are unwilling to assume a

particular probability distribution, the likelihood ratio principle cannot be used. This could

lead to the use of the nonparametric test procedures discussed in Chapter 15.

9-5.2 Small-Sample Tests on a Proportion (CD Only)

Tests on a proportion when the sample size nis small are based on the binomial distribution,

not the normal approximation to the binomial. To illustrate, suppose we wish to test

Let Xbe the number of successes in the sample. A lower-tail rejection region would be used.

That is, we would reject H 0 if xc, where cis the critical value. When H 0 is true, Xhas a

binomial distribution with parameters nand p 0 ; therefore,

B 1 c; n, p 02

P 3 Xc when X is Bin 1 n, p 024

P 1 Type I error 2 P 1 reject H 0 when H 0 is true 2

H 1 : pp 0

H 0 : p p 0

t^2 2,n 

1

t

t2,n
1 and tt2,n
1

 c

1

1  3 t^2  1 n 

124

s

n 2

c

1 x

 022

s^2 n

d t^2

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