8.6Hypothesis Tests in Bernoulli Populations 327
8.6.1 Testing the Equality of Parameters in Two
Bernoulli Populations
Suppose there are two distinct methods for producing a certain type of transistor; and
suppose that transistors produced by the first method will, independently, be defective
with probabilityp 1 , with the corresponding probability beingp 2 for those produced by
the second method. To test the hypothesis thatp 1 =p 2 , a sample ofn 1 transistors is
produced using method 1 andn 2 using method 2.
LetX 1 denote the number of defective transistors obtained from the first sample andX 2
for the second. Thus,X 1 andX 2 are independent binomial random variables with respective
parameters (n 1 ,p 1 ) and (n 2 ,p 2 ). Suppose thatX 1 +X 2 =kand so there have been a total
ofkdefectives. Now, ifH 0 is true, then each of then 1 +n 2 transistors produced will have
the same probability of being defective, and so the determination of thekdefectives will
have the same distribution as a random selection of a sample of sizekfrom a population
ofn 1 +n 2 items of whichn 1 are white andn 2 are black. In other words, given a total of
kdefectives, the conditional distribution of the number of defective transistors obtained
from method 1 will, whenH 0 is true, have the following hypergeometric distribution*:
PH 0 {X 1 =i|X 1 +X 2 =k}=(
n 1
i)(
n 2
k−i)(
n 1 +n 2
k) , i=0, 1,...,k (8.6.1)Now, in testing
H 0 :p 1 =p 2 versus H 1 :p 1 =p 2it seems reasonable to reject the null hypothesis when the proportion of defective transistors
produced by method 1 is much different than the proportion of defectives obtained under
method 2. Therefore, if there is a total ofkdefectives, then we would expect, whenH 0
is true, thatX 1 /n 1 (the proportion of defective transistors produced by method 1) would
be close to (k−X 1 )/n 2 (the proportion of defective transistors produced by method 2).
BecauseX 1 /n 1 and (k−X 1 )/n 2 will be farthest apart whenX 1 is either very small or very
large, it thus seems that a reasonable significance levelαtest of Equation 8.6.1 is as follows.
IfX 1 +X 2 =k, then one should
reject H 0 if either P{X≤x 1 }≤α/2 or P{X≥x 1 }≤α/2
accept H 0 otherwise* See Example 5.3b for a formal verification of Equation 8.6.1.