Introductory Biostatistics

(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