Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.5. Introduction to Hypothesis Testing 269

Over all critical regions of sizeα, we want to consider critical regions that have
lower probabilities of Type II error. We also can look at the complement of a Type
II error, namely, rejectingH 0 whenH 1 is true, which is a correct decision, as marked
in Table 4.5.1. Since we desire to maximize the probability of this latter decision,
we want the probability of it to be as large as possible. That is, forθ∈ω 1 ,wewant
to maximize
1 −Pθ[Type II Error] =Pθ[(X 1 ,...,Xn)∈C].
The probability on the right side of this equation is called thepowerof the test
atθ. It is the probability that the test detects the alternativeθwhenθ∈ω 1 is
the true parameter. So minimizing the probability of Type II error is equivalent to
maximizing power.
We define thepower functionof a critical region to be

γC(θ)=Pθ[(X 1 ,...,Xn)∈C]; θ∈ω 1. (4.5.5)

Hence, given two critical regionsC 1 andC 2 , which are both of sizeα,C 1 is better
thanC 2 ifγC 1 (θ)≥γC 2 (θ) for allθ∈ω 1. In Chapter 8, we obtain optimal critical
regions for specific situations. In this section, we want to illustrate these concepts
of hypothesis testing with several examples.

Example 4.5.2(Test for a Binomial Proportion of Success).LetXbe a Bernoulli
random variable with probability of successp. Suppose we want to test, at sizeα,

H 0 : p=p 0 versusH 1 : p<p 0 , (4.5.6)

wherep 0 is specified. As an illustration, suppose “success” is dying from a certain

disease andp 0 is the probability of dying with some standard treatment. A new

treatment is used on several (randomly chosen) patients, and it is hoped that the

probability of dying under this new treatment is less thanp 0 .LetX 1 ,...,Xnbe

a random sample from the distribution ofXand letS=

∑n

i=1Xibe the total

number of successes in the sample. An intuitive decision rule (critical region) is

RejectH 0 in favor ofH 1 ifS≤k, (4.5.7)

wherekis such thatα=PH 0 [S≤k]. SinceShas ab(n, p 0 ) distribution underH 0 ,
kis determined byα=Pp 0 [S≤k]. Because the binomial distribution is discrete,
however, it is likely that there is no integerkthat solves this equation. For example,
supposen= 20,p 0 =0.7, andα=0.15. Then underH 0 ,Shas a binomialb(20, 0 .7)
distribution. Hence, computationally,PH 0 [S≤11] =pbinom(11,20,0.7)=0. 1133
andPH 0 [S≤12] =pbinom(12,20,0.7)=0.2277. Hence, erring on the conservative
side, we would probably choosekto be 11 andα=0.1133. Asnincreases, this is
less of a problem; see, also, the later discussion onp-values. In general, the power
of the test for the hypotheses (4.5.6) is

γ(p)=Pp[S≤k],p<p 0. (4.5.8)

The curve labeled Test 1 in Figure 4.5.1 is the power function for the casen= 20,

p 0 =0.7, andα=0.1133. Notice that the power function is decreasing. The