278 Some Elementary Statistical Inferencesthat is, we acceptH 0 at significance levelαif and only ifμ 0 is in the (1−α)100%
confidence interval forμ. Equivalently, we rejectH 0 at significance levelαif and
only ifμ 0 is not in the (1−α)100% confidence interval forμ. This is true for all
the two-sided tests and hypotheses discussed in this text. There is also a similar
relationship between one-sided tests and one-sided confidence intervals.
Once we recognize this relationship between confidence intervals and tests of
hypothesis, we can use all those statistics that we used to construct confidence
intervals to test hypotheses, not only against two-sided alternatives but one-sided
ones as well. Without listing all of these in a table, we present enough of them so
that the principle can be understood.
Example 4.6.2. Let independent random samples be taken fromN(μ 1 ,σ^2 )and
N(μ 2 ,σ^2 ), respectively. Say these have the respective sample characteristicsn 1 ,
X,S 12 andn 2 ,Y,S^22 .Letn=n 1 +n 2 denote the combined sample size and let
S^2 p=[(n 1 −1)S 12 +(n 2 −1)S 22 ]/(n−2), (4.2.11), be the pooled estimator of the
common variance. Atα=0.05, rejectH 0 :μ 1 =μ 2 and accept the one-sided
alternativeH 1 :μ 1 >μ 2 if
T=X−Y− 0
Sp√
1
n 1 +1
n 2≥t. 05 ,n− 2 ,because, underH 0 :μ 1 =μ 2 ,Thas at(n−2)-distribution. A rigorous development
of this test is given in Example 8.3.1.
Example 4.6.3.SayXisb(1,p). Consider testingH 0 :p=p 0 againstH 1 :p<p 0.
LetX 1 ...,Xnbe a random sample from the distribution ofXand let̂p=X.To
testH 0 versusH 1 , we use either
Z 1 =p̂−p 0
√
p 0 (1−p 0 )/n≤c or Z 2 =p̂−p 0
√
p̂(1−p̂)/n≤c.Ifnis large, bothZ 1 andZ 2 have approximate standard normal distributions pro-
vided thatH 0 :p=p 0 is true. Hence, ifcis set at− 1 .645, then the approximate
significance level isα=0.05. Some statisticians useZ 1 and othersZ 2 .Wedo
not have strong preferences one way or the other because the two methods provide
about the same numerical results. As one might suspect, usingZ 1 provides better
probabilities for power calculations if the truepis close top 0 , whileZ 2 is better
ifH 0 is clearly false. However, with a two-sided alternative hypothesis,Z 2 does
provide a better relationship with the confidence interval forp.Thatis,|Z 2 |<zα/ 2
is equivalent top 0 being in the interval from
p̂−zα/ 2√
̂p(1−̂p)
nto p̂+zα/ 2√
p̂(1−p̂)
n,which is the interval that provides a (1−α)100% approximate confidence interval
forpas considered in Section 4.2.
In closing this section, we introduce the concept ofrandomized tests.