Inferential Statistics 381
P(δ(X) = d 1 ), computed for all values θ in the set Θ 1 corresponding to the
alternative hypothesis.
Recall that we can either obtain d 0 or d 1 as a test result, no matter what
the value of the parameter may truly be. Since d 0 and d 1 are mutually exclu-
sive, we have the relation
P(δ(X) = d 0 ) + P(δ(X) = d 1 ) = 1Now, for any parameter value θ from Θ 1 , this means that the power of
the test and the probability of committing a type II error, PII(δ(X)), add up
to one. We illustrate this in Figure C.5. The dashed lines indicate the prob-
ability PII(δ(X)), respectively, at the corresponding parameter values θ, while
the dash-dotted lines represent the power for given θ ∈ Θ 1. As we can see,
the power gradually takes over much of the probability mass from the type
II error probability the greater the parameter values.
Suppose of all the tests with significance level α, we had one δ*, which
always had greater power than any of the others. Then it would be reason-
able to prefer this test to all the others since we have the smallest chance of
incurring a type II error. This leads to the following concept.
figURe C.4 The Solid Line of the Probability P(δ(X) = d 1 ), over the Set Θ 1 ,
Indicates the Power of the Test δ
Θ 0 Θ 1 θP(δ(X) = d 1 )