Introduction to Probability and Statistics for Engineers and Scientists

8.3Tests Concerning the Mean of a Normal Population 305

TABLE 8.1 X 1 ,...,XnIs a Sample from aN(μ,σ^2 )Populationσ^2 Is KnownX=

∑n

i= 1

Xi/n

Significance

H 0 H 1 Test StatisticTS LevelαTest p-Value ifTS=t

μ=μ 0 μ=μ 0 √n(X−μ 0 )/σ Reject if|TS|>zα/2 2 P{Z≥|t|}
μ≤μ 0 μ>μ 0 √n(X−μ 0 )/σ Reject ifTS>zα P{Z≥t}
μ≥μ 0 μ<μ 0

√

n(X−μ 0 )/σ Reject ifTS<−zα P{Z≤t}

Z is a standard normal random variable.

(b) A Remark on Robustness A test that performs well even when the underlying
assumptions on which it is based are violated is said to berobust. For instance, the tests
of Sections 8.3.1 and 8.3.1.1 were derived under the assumption that the underlying
population distribution is normal with known varianceσ^2. However, in deriving these
tests, this assumption was used only to conclude thatX also has a normal distribution.
But, by the central limit theorem, it follows that for a reasonably large sample size,Xwill
approximatelyhaveanormaldistributionnomatterwhattheunderlyingdistribution. Thus
we can conclude that these tests will be relatively robust for any population distribution
with varianceσ^2.
Table 8.1 summarizes the tests of this subsection.

8.3.2 Case of Unknown Variance: Thet-Test.............................

Up to now we have supposed that the only unknown parameter of the normal population
distribution is its mean. However, the more common situation is one where the meanμ
and varianceσ^2 are both unknown. Let us suppose this to be the case and again consider a
test of the hypothesis that the mean is equal to some specified valueμ 0. That is, consider
a test of

H 0 :μ=μ 0

versus the alternative

H 1 :μ=μ 0

It should be noted that the null hypothesis is not a simple hypothesis since it does not
specify the value ofσ^2.
As before, it seems reasonable to rejectH 0 when the sample meanXis far fromμ 0.
However, how far away it need be to justify rejection will depend on the varianceσ^2.
Recall that when the value ofσ^2 was known, the test called for rejectingH 0 when|X−μ 0 |
exceededzα/2σ/

√

nor, equivalently, when

∣∣

∣∣

∣

X−μ 0

σ/

√

n

∣∣

∣∣

∣

>zα/2