270 Some Elementary Statistical Inferencespower is higher to detect the alternativep=0.2thanp=0.6. In Section 8.2, we
prove in general the monotonicity of the power function for binomial tests of these
hypotheses. Using this monotonicity, we extend our test to the more general null
hypothesisH 0 : p≥p 0 rather than simplyH 0 : p=p 0. Using the same decision
rule as we used for the hypotheses (4.5.6), the definition of the size of a test (4.5.4),
and the monotonicity of the power curve, we have
max
p≥p 0
Pp[S≤k]=Pp 0 [S≤k]=α,i.e., the same size as for the original null hypothesis.p
0.4 0.5 0.7 0.80.80.40.2(p)Test 1: size = 0.113Test 2: size = 0.227Figure 4.5.1:Power curves for tests 1 and 2; see Example 4.5.2.Denote by Test 1 the test for the situation withn= 20,p 0 =0.70, and size
α=0.1133. Suppose we have a second test (Test 2) with an increased size. How
does the power function of Test 2 compare to Test 1? As an example, suppose
for Test 2, we selectα =0.2277. Hence, for Test 2, we rejectH 0 ifS ≤12.
Figure 4.5.1 displays the resulting power function. Note that while Test 2 has a
higher probability of committing a Type I error, it also has a higher power at each
alternativep< 0 .7. Exercise 4.5.7 shows that this is true for these binomial tests.
It is true in general; that is, if the size of the test increases, power does too. For
this example, the R functionbinpower.r, found at the site listed in the Preface,
produces a version of Figure 4.5.1.
Remark 4.5.1(Nomenclature).Since in Example 4.5.2, the first null hypothesis
H 0 : p=p 0 completely specifies the underlying distribution, it is called asimple
hypothesis. Most hypotheses, such asH 1 : p<p 0 ,arecompositehypotheses,
because they are composed of many simple hypotheses and, hence, do not completely
specify the distribution.
As we study more and more statistics, we discover that often other names are
used for the size,α, of the critical region. Frequently,αis also called thesignifi-