Introduction to Probability and Statistics for Engineers and Scientists

292 Chapter 8:Hypothesis Testing

8.2Significance Levels

Consider a population having distributionFθ, whereθis unknown, and suppose we want
to test a specific hypothesis aboutθ. We shall denote this hypothesis byH 0 and call it
thenull hypothesis. For example, ifFθis a normal distribution function with meanθand
variance equal to 1, then two possible null hypotheses aboutθare

(a)H 0 :θ= 1

(b)H 0 :θ≤ 1

Thus the first of these hypotheses states that the population is normal with mean 1 and
variance 1, whereas the second states that it is normal with variance 1 and a mean less than
or equal to 1. Note that the null hypothesis in (a), when true, completely specifies the
population distribution; whereas the null hypothesis in (b) does not. A hypothesis that,
when true, completely specifies the population distribution is called asimplehypothesis;
one that does not is called acompositehypothesis.
Suppose now that in order to test a specific null hypothesisH 0 , a population sample
of sizen— sayX 1 ,...,Xn— is to be observed. Based on thesenvalues, we must decide
whether or not to acceptH 0. A test forH 0 can be specified by defining a regionCin
n-dimensional space with the proviso that the hypothesis is to be rejected if the random
sampleX 1 ,...,Xnturns out to lie inCand accepted otherwise. The regionCis called the
critical region. In other words, the statistical test determined by the critical regionCis the
one that

accepts H 0 if (X 1 ,X 2 ,...,Xn)∈C

and

rejects H 0 if (X 1 ,...,Xn)∈C

For instance, a common test of the hypothesis thatθ, the mean of a normal population
with variance 1, is equal to 1 has a critical region given by

C=







(X 1 ,...,Xn):

∣

∣∣

∣

∣∣

∣∣

∑n

i= 1

Xi

n

− 1

∣

∣∣

∣

∣∣

∣∣

>

1.96

√

n







(8.2.1)

Thus, this test calls for rejection of the null hypothesis thatθ=1 when the sample average
differs from 1 by more than 1.96 divided by the square root of the sample size.
It is important to note when developing a procedure for testing a given null hypothesis
H 0 that, in any test, two different types of errors can result. The first of these, called atype
I error, is said to result if the test incorrectly calls for rejectingH 0 when it is indeed correct.
The second, called atype II error, results if the test calls for acceptingH 0 when it is false.