Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

276 Some Elementary Statistical Inferences

Example 4.6.1(Large Sample Two-Sided Test for the Mean).In order to see how
to construct a test for a two-sided alternative, reconsider Example 4.5.3, where we
constructed a large sample one-sided test for the mean of a random variable. As
in Example 4.5.3, letXbe a random variable with meanμand finite varianceσ^2.
Here, though, we want to test

H 0 : μ=μ 0 versusH 1 : μ =μ 0 , (4.6.1)

whereμ 0 is specified. LetX 1 ,...,Xnbe a random sample from the distribution of
Xand denote the sample mean and variance byXandS^2 , respectively. For the
one-sided test, we rejectedH 0 ifXwas too large; hence, for the hypotheses (4.6.1),
we use the decision rule

RejectH 0 in favor ofH 1 ifX≤horX≥k, (4.6.2)

wherehandkare such thatα=PH 0 [X≤horX≥k]. Clearly,h<k; hence, we
have
α=PH 0 [X≤horX≥k]=PH 0 [X≤h]+PH 0 [X≥k].

Since, at least for large samples, the distribution ofXis symmetrically distributed
aboutμ 0 , underH 0 , an intuitive rule is to divideαequally between the two terms
on the right side of the above expression; that is,handkare chosen by

PH 0 [X≤h]=α/2andPH 0 [X≥k]=α/ 2. (4.6.3)

From Theorem 4.2.1, it follows that (X−μ 0 )/(S/

√
n) is approximatelyN(0,1).
This and (4.6.3) lead to the approximate decision rule

RejectH 0 in favor ofH 1 if

∣

∣

∣XS/−√μn^0

∣

∣

∣≥zα/ 2. (4.6.4)

Upon substitutingσforS, it readily follows that the approximate power function

is

γ(μ)=Pμ(X≤μ 0 −zα/ 2 σ/

√

n)+Pμ(X≥μ 0 +zα/ 2 σ/

√

n)

=Φ

(√

n(μ 0 −μ)

σ

−zα/ 2

)

+1−Φ

(√

n(μ 0 −μ)

σ

+zα/ 2

)

,(4.6.5)

where Φ(z) is the cdf of a standard normal random variable; see (3.4.9). So if we

have some reasonable idea of whatσequals, we can compute the approximate power

function. Note that the derivative of the power function is

γ′(μ)=

√

n

σ

[

φ

(√

n(μ 0 −μ)

σ

+zα/ 2

)

−φ

(√

n(μ 0 −μ)

σ

−zα/ 2

)]

, (4.6.6)

whereφ(z) is the pdf of a standard normal random variable. Then we can show that
γ(μ) has a critical value atμ 0 which is the minimum; see Exercise 4.6.2. Further,
γ(μ) is strictly decreasing forμ<μ 0 and strictly increasing forμ>μ 0.