Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

268 Some Elementary Statistical Inferences

Table 4.5.1: 2 ×2 Decision Table for a Hypothesis Test

True State of Nature

Decision H 0 is True H 1 is True

RejectH 0 Type I Error Correct Decision

AcceptH 0 Correct Decision Type II Error

E(Yi)>E(Zi); i.e., on average the cross-fertilized plants are taller than the self-

fertilized plants. Under this model, our hypotheses are

H 0 :μ=0versusH 1 : μ> 0. (4.5.2)

Hence,ω 0 ={ 0 }represents no difference in the treatments, whileω 1 =(0,∞)
represents that the mean height of cross-fertilizedZea maysexceeds the mean height
of self-fertilizedZea mays.

To complete the testing structure for the general problem described at the be-
ginning of this section, we need to discuss decision rules. Recall thatX 1 ,...,Xn
is a random sample from the distribution of a random variableX that has den-
sityf(x;θ), whereθ∈Ω. Consider testing the hypothesesH 0 : θ∈ω 0 versus
H 1 : θ∈ω 1 ,whereω 0 ∪ω 1 = Ω. Denote the space of the sample byD;thatis,
D=space{(X 1 ,...,Xn)}.AtestofH 0 versusH 1 is based on a subsetCofD.
This setCis called thecritical regionand its corresponding decision rule (test)
is

RejectH 0 (AcceptH 1 )if(X 1 ,...,Xn)∈C (4.5.3)

RetainH 0 (RejectH 1 )if(X 1 ,...,Xn)∈Cc.

For a given critical region, the 2×2 decision table as shown in Table 4.5.1,
summarizes the results of the hypothesis test in terms of the true state of nature.
Besides the correct decisions, two errors can occur. AType Ierror occurs ifH 0 is
rejected when it is true, while aType IIerror occurs ifH 0 is accepted whenH 1 is
true.
The goal, of course, is to select a critical region from all possible critical regions
which minimizes the probabilities of these errors. In general, this is not possible.
The probabilities of these errors often have a seesaw effect. This can be seen imme-
diately in an extreme case. Simply letC=φ. With this critical region, we would
never rejectH 0 , so the probability of Type I error would be 0, but the probability of
Type II error is 1. Often we consider Type I error to be the worse of the two errors.
We then proceed by selecting critical regions that bound the probability of Type I
error and then among these critical regions we try to select one that minimizes the
probability of Type II error.

Definition 4.5.1. We say a critical regionCis ofsizeαif

α=max

θ∈ω 0

Pθ[(X 1 ,...,Xn)∈C]. (4.5.4)