Introduction to Probability and Statistics for Engineers and Scientists

306 Chapter 8:Hypothesis Testing

Now whenσ^2 is no longer known, it seems reasonable to estimate it by

S^2 =

∑n

i= 1

(Xi−X)^2

n− 1

and then to rejectH 0 when
∣
∣∣
∣∣

X−μ 0

S/

√

n

∣

∣∣

∣∣

is large.
To determine how large a value of the statistic
∣
∣∣
∣∣

√

n(X−μ 0 )

S

∣

∣∣

∣∣

to require for rejection, in order that the resulting test have significance levelα, we must
determine the probability distribution of this statistic whenH 0 is true. However, as shown
in Section 6.5, the statisticT, defined by

T=

√

n(X−μ 0 )

S

has, whenμ=μ 0 ,at-distribution withn−1 degrees of freedom. Hence,

Pμ 0

{

−tα/2,n− 1 ≤

√

n(X−μ 0 )

S

≤tα/2,n− 1

}

= 1 −α (8.3.11)

wheretα/2,n− 1 is the 100α/2 upper percentile value of thet-distribution withn−1 degrees
of freedom. (That is,P{Tn− 1 ≥tα/2,n− 1 }=P{Tn− 1 ≤−tα/2,n− 1 }=α/2 whenTn− 1
has at-distribution withn−1 degrees of freedom.) From Equation 8.3.11 we see that the
appropriate significance levelαtest of

H 0 :μ=μ 0 versus H 1 :μ=μ 0

is, whenσ^2 is unknown, to

accept H 0 if

∣∣

∣

∣∣

√

n(X−μ 0 )

S

∣∣

∣

∣∣≤tα/2,n− 1

reject H 0 if

∣∣

∣∣

∣

√

n(X−μ 0 )

S

∣∣

∣∣

∣

>tα/2,n− 1

(8.3.12)