Introduction to Probability and Statistics for Engineers and Scientists

14.4A Two-Sample Problem 599

Hence, by the equivalence of the gamma and chi-square distribution it follows that

2

θ 1

∑n

i= 1

Xi∼χ 22 n

2

θ 2

∑m

i= 1

Yi∼χ 22 m

Hence, it follows from the definition of theF-distribution that

(

2

θ 1

∑n

i= 1

Xi

)

(^2 n

2

θ 2

∑m

i= 1

Yi

)

2 m

∼Fn,m

That is, ifXandYare the two sample means, respectively, then

θ 2 X

θ 1 Y

has anF-distribution withnandmdegrees of freedom

Hence, when the hypothesisθ 1 =θ 2 is true, we see thatX/Yhas anF-distribution with
nandmdegrees of freedom. This suggests the following test of the hypothesis that
θ 1 =θ 2.

Test: H 0 :θ 1 =θ 2 vs. alternativeH 1 :θ 1 =θ 2

Step 1: Choose a significance levelα.

Step 2: Determine the value of the test statisticX/Y— say its value isv.

Step 3: ComputeP{F≤v}whereF∼Fn,m. If this probability is either less

thanα/2 (which occurs whenXis significantly less thanY) or greater

than 1−α/2 (which occurs whenXis significantly greater thanY) then

the hypothesis is rejected.

In other words, thep-value of the test data is given by

p-value=2 min(P{F≤v},1−P{F≤v})

EXAMPLE 14.4a Test the hypothesis, at the 5 percent level of significance, that the lifetimes
of items produced at two given plants have the same exponential life distribution if a sample
of size 10 from the first plant has a total lifetime of 420 hours whereas a sample of 15 from
the second plant has a total lifetime of 510 hours.