Applied Statistics and Probability for Engineers

10-6.2 Small-Sample Test for H 0 : p 1 p 2 (CD Only)

10-6.3 -Error and Choice of Sample Size

The computation of the -error for the large-sample test of H 0 : p 1 p 2 is somewhat more

involved than in the single-sample case. The problem is that the denominator of the test

statistic Z 0 is an estimate of the standard deviation of under the assumption that p 1 

p 2 p. When H 0 : p 1 p 2 is false, the standard deviation of is

(^) Pˆ 1 Pˆ 2  (10-34)
B
p 111 p 12
n 1
p 211 p 22
n 2
Pˆ 1 Pˆ 2
Pˆ 1 Pˆ 2
364 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES
If the alternative hypothesis is two sided, the -error is
  c (10-35)
z 22 pq (^11) n 1
1 n 22  1 p 1 p 22
(^) Pˆ 1 Pˆ 2 d
 c
z 22 pq (^11) n 1
1 n 22  1 p 1 p 22
(^) Pˆ 1 Pˆ 2 d
If the alternative hypothesis is H 1 : p 1 p 2 ,
(10-36)
and if the alternative hypothesis is H 1 : p 1 p 2 ,
 1  c (10-37)
z 2 pq (^11) n 1
1 n 22  1 p 1 p 22
(^) Pˆ 1 Pˆ 2
d
 c
z 2 pq (^11) n 1
1 n 22  1 p 1 p 22
(^) Pˆ 1 Pˆ 2 d
where
and (^) Pˆ 1 Pˆ 2 is given by Equation 10-34.
p
n 1 p 1
n 2 p 2
n 1
n 2

and q

n 111 p 12

 n 211 p 22

n 1

 n 2

For a specified pair of values p 1 and p 2 , we can find the sample sizes n 1 n 2 nrequired to

give the test of size that has specified type II error.

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