4.5. Introduction to Hypothesis Testing 271cance levelof the test associated with that critical region. Moreover, sometimes
αis called the “maximum of probabilities of committing an error of Type I” and
the“maximumofthepowerofthetestwhenH 0 is true.” It is disconcerting to the
student to discover that there are so many names for the same thing. However, all
of them are used in the statistical literature, and we feel obligated to point out this
fact.
The test in the last example is based on the exact distribution of its test statistic,
i.e., the binomial distribution. Often we cannot obtain the distribution of the test
statistic in closed form. As with approximate confidence intervals, however, we can
frequently appeal to the Central Limit Theorem to obtain an approximate test; see
Theorem 4.2.1. Such is the case for the next example.
Example 4.5.3(Large Sample Test for the Mean). LetXbe a random variable
with meanμand finite varianceσ^2. We want to test the hypotheses
H 0 : μ=μ 0 versusH 1 : μ>μ 0 , (4.5.9)
whereμ 0 is specified. To illustrate, supposeμ 0 is the mean level on a standardized
test of students who have been taught a course by a standard method of teaching.
Suppose it is hoped that a new method that incorporates computers has a mean
levelμ>μ 0 ,whereμ=E(X)andXis the score of a student taught by the new
method. This conjecture is tested by havingnstudents (randomly selected) taught
under this new method.
LetX 1 ,...,Xnbe a random sample from the distribution ofXand denote the
sample mean and variance byXandS^2 , respectively. BecauseXis an unbiased
estimate ofμ, an intuitive decision rule is given by
RejectH 0 in favor ofH 1 ifXis much larger thanμ 0. (4.5.10)
In general, the distribution of the sample mean cannot be obtained in closed form.
In Example 4.5.4, under the strong assumption of normality for the distribution of
X, we obtain an exact test. For now, the Central Limit Theorem (Theorem 4.2.1)
shows that the distribution of (X−μ)/(S/
√
n) is approximatelyN(0,1). Using
this, we obtain a test with an approximate sizeα, with the decision rule
RejectH 0 in favor ofH 1 ifXS/−√μn^0 ≥zα. (4.5.11)The test is intuitive. To rejectH 0 ,X must exceedμ 0 by at leastzαS/
√
n.To
approximate the power function of the test, we use the Central Limit Theorem.
Upon substitutingσforS, it readily follows that the approximate power function
is
γ(μ)=Pμ(X≥μ 0 +zασ/√
n)= Pμ(
X−μ
σ/√
n≥
μ 0 −μ
σ/√
n+zα)≈ 1 −Φ(
zα+√
n(μ 0 −μ)
σ)=Φ(
−zα−√
n(μ 0 −μ)
σ). (4.5.12)