(a) AlphaðaÞis defined as the probability of wrongly rejecting a true null
hypothesis, that is,
a¼Prðpa 0 : 20 ;given thatp¼ 0 : 25 Þ
Sincen¼100 is large enough for the central limit theorem to apply, the sam-
pling distribution ofpis approximately normal with mean and variance, under
H 0 , given by
mp¼p
¼ 0 : 25
sp^2 ¼
pð 1 pÞ
n
¼ð 0 : 043 Þ^2
respectively. Therefore, for this decision-making rule,
a¼Pr za
0 : 20 0 : 25
0 : 043
¼Prðza 1 : 16 Þ
¼ 0 :123 or 12:3%
Of course, we can make this smaller (as small as we wish) by changing the
decision-making rule; however, that action will increase the value ofb(or the
probability of a type II error).
(b) Suppose that the truth is
HA:p¼ 0 : 15
BetaðbÞis defined as the probability of not rejecting a falseH 0 , that is,
b¼Prðp> 0 : 20 ;knowing thatp¼ 0 : 15 Þ
Again, an application of the central limit theorem indicates that the sampling
distribution ofpis approximately normal with mean
mp¼ 0 : 15
and variance
sp^2 ¼
ð 0 : 15 Þð 0 : 85 Þ
100
¼ð 0 : 036 Þ^2
BASIC CONCEPTS 193