Introduction to Probability and Statistics for Engineers and Scientists

294 Chapter 8:Hypothesis Testing

SinceX=

∑n
i= 1 Xi/nis a natural point estimator ofμ, it seems reasonable to accept
H 0 ifXis not too far fromμ 0. That is, the critical region of the test would be of the form

C={X 1 ,...,Xn:|X−μ 0 |>c} (8.3.1)

for some suitably chosen valuec.
If we desire that the test has significance levelα, then we must determine the critical
valuecin Equation 8.3.1 that will make the type I error equal toα. That is,cmust be
such that

Pμ 0 {|X−μ 0 |>c}=α (8.3.2)

where we writePμ 0 to mean that the preceding probability is to be computed under the
assumption thatμ=μ 0. However, whenμ=μ 0 ,Xwill be normally distributed with
meanμ 0 and varianceσ^2 /nand soZ, defined by

Z≡

X−μ 0

σ/

√

n

will have a standard normal distribution. Now Equation 8.3.2 is equivalent to

P

{

|Z|>

c

√

n

σ

}

=α

or, equivalently,

2 P

{

Z>

c

√

n

σ

}

=α

whereZis a standard normal random variable. However, we know that

P{Z>zα/2}=α/2

and so
c

√

n

σ

=zα/2

or

c=

zα/2σ

√

n

Thus, the significance levelαtest is to rejectH 0 if|X−μ 0 |>zα/2σ/

√
nand accept
otherwise; or, equivalently, to

reject H 0 if

√

n

σ

|X−μ 0 |>zα/2

accept H 0 if

√

n

σ

|X−μ 0 |≤zα/2

(8.3.3)