Introduction to Probability and Statistics for Engineers and Scientists

8.5Hypothesis Tests Concerning the Variance of a Normal Population 321

8.5Hypothesis Tests Concerning the Variance of a Normal Population

LetX 1 ,...,Xndenote a sample from a normal population having unknown meanμand
unknown varianceσ^2 , and suppose we desire to test the hypothesis

H 0 :σ^2 =σ 02

versus the alternative

H 1 :σ^2 =σ 02

for some specified valueσ 02.
To obtain a test, recall (as was shown in Section 6.5) that (n−1)S^2 /σ^2 has a chi-square
distribution withn−1 degrees of freedom. Hence, whenH 0 is true

(n−1)S^2

σ 02

∼χn^2 − 1

and so

PH 0

{

χ 12 −α/2,n− 1 ≤

(n−1)S^2

σ 02

≤χα^2 /2,n− 1

}

= 1 −α

Therefore, a significance levelαtest is to

accept H 0 if χ 12 −α/2,n− 1 ≤

(n−1)S^2

σ 02

≤χα^2 /2,n− 1

reject H 0 otherwise

The preceding test can be implemented by first computing the value of the test statistic
(n−1)S^2 /σ 02 — call itc. Then compute the probability that a chi-square random variable
withn−1 degrees of freedom would be (a) less than and (b) greater thanc. If either of
these probabilities is less thanα/2, then the hypothesis is rejected. In other words, the
p-value of the test data is

p-value=2 min(P{χn^2 − 1 <c},1−P{χn^2 − 1 <c})

The quantityP{χn^2 − 1 < c}can be obtained from Program 5.8.1.A. Thep-value for
a one-sided test is similarly obtained.

EXAMPLE 8.5a A machine that automatically controls the amount of ribbon on a tape has
recently been installed. This machine will be judged to be effective if the standard deviation
σof the amount of ribbon on a tape is less than .15 cm. If a sample of 20 tapes yields
a sample variance ofS^2 =.025 cm^2 , are we justified in concluding that the machine is
ineffective?