Introduction to Probability and Statistics for Engineers and Scientists

8.3Tests Concerning the Mean of a Normal Population 309

against the one-sided alternative

H 1 :μ>μ 0

The significance levelαtest is to

accept H 0 if

√

n(X−μ 0 )

S

≤tα,n− 1

reject H 0 if

√

n(X−μ 0 )

S

>tα,n− 1

(8.3.13)

If

√
n(X−μ 0 )/S =v, then thep-value of the test is the probability that at-random
variable withn−1 degrees of freedom would be at least as large asv.
The significance levelαtest of

H 0 :μ=μ 0 (orH 0 :μ≥μ 0 )

versus the alternative

H 1 :μ<μ 0

is to

accept H 0 if

√

n(X−μ 0 )

S

≥−tα,n− 1

reject H 0 if

√

n(X−μ 0 )

S

<−tα,n− 1

Thep-value of this test is the probability that at-random variable withn−1 degrees of
freedom would be less than or equal to the observed value of

√

n(X−μ 0 )/S.

EXAMPLE 8.3i The manufacturer of a new fiberglass tire claims that its average life will be
at least 40,000 miles. To verify this claim a sample of 12 tires is tested, with their lifetimes
(in 1,000s of miles) being as follows:

Tire 1 2 3 4 5 6 7 8 9 10 11 12

Life 36.1 40.2 33.8 38.5 42 35.8 37 41 36.8 37.2 33 36

Test the manufacturer’s claim at the 5 percent level of significance.

SOLUTION To determine whether the foregoing data are consistent with the hypothesis
that the mean life is at least 40,000 miles, we will test

H 0 :μ≥40,000 versus H 1 :μ<40,000