Introduction to Probability and Statistics for Engineers and Scientists

8.3Tests Concerning the Mean of a Normal Population 301

Hence, the test of the hypothesis 8.3.8 is to rejectH 0 ifX−μ 0 >zασ/

√
n, and accept
otherwise; or, equivalently, to

accept H 0 if

√

n

σ

(X−μ 0 )≤zα

reject H 0 if

√

n

σ

(X−μ 0 )>zα

(8.3.10)

This is called aone-sidedcritical region (since it calls for rejection only whenXis large).
Correspondingly, the hypothesis testing problem

H 0 :μ=μ 0

H 1 :μ>μ 0

is called a one-sided testing problem (in contrast to thetwo-sidedproblem that results when
the alternative hypothesis isH 1 :μ=μ 0 ).
To compute thep-value in the one-sided test, Equation 8.3.10, we first use the data
to determine the value of the statistic

√
n(X−μ 0 )/σ. Thep-value is then equal to the
probability that a standard normal would be at least as large as this value.

EXAMPLE 8.3e Suppose in Example 8.3a that we know in advance that the signal value is
at least as large as 8. What can be concluded in this case?

SOLUTION To see if the data are consistent with the hypothesis that the mean is 8, we test

H 0 :μ= 8

against the one-sided alternative

H 1 :μ> 8

The value of the test statistic is

√

n(X−μ 0 )/σ=

√
5(9.5−8)/2=1.68, and thep-value
is the probability that a standard normal would exceed 1.68, namely,

p-value= 1 − (1.68)=.0465

Since the test would call for rejection at all significance levels greater than or equal to .0465,
it would, for instance, reject the null hypothesis at theα=.05 level of significance. ■

The operating characteristic function of the one-sided test, Equation 8.3.10,

β(μ)=Pμ{acceptingH 0 }