Applied Statistics and Probability for Engineers

For a one-sided alternative, replace in Equation 10-38 by z.

10-6.4 Confidence Interval for p 1 p 2

The confidence interval for p 1 p 2 can be found directly, since we know that

is a standard normal random variable. Thus P(z 2 Zz 2 )  1 , so we can substi-

tute for Zin this last expression and use an approach similar to the one employed previously

to find an approximate 100(1 )% two-sided confidence interval for p 1 p 2.

Z

Pˆ 1 Pˆ 2  1 p 1 p 22

B

p 111 p 12

n 1

p 211 p 22

n 2

z 2

10-6 INFERENCE ON TWO POPULATION PROPORTIONS 365

For the two-sided alternative, the common sample size is

(10-38)

where q 1  1 p 1 and q 2  1 p 2.

n

3 z 211 p 1

 p 221 q 1

 q 22  2

 z 1 p 1 q 1

 p 2 q 242

1 p 1 p 222

If and are the sample proportions of observation in two independent random

samples of sizes n 1 and n 2 that belong to a class of interest, an approximate two-

sided 100(1)% confidence interval on the difference in the true proportions

p 1 p 2 is

(10-39)

where z 2 is the upper 2 percentage point of the standard normal distribution.

p 1 p 2 pˆ 1 pˆ 2

 z 2

B

pˆ 111 pˆ 12

n 1

pˆ 211 pˆ 22

n 2

pˆ 1 pˆ 2 z 2

B

pˆ 111 pˆ 12

n 1

pˆ 211 pˆ 22

n 2

pˆ 1 pˆ 2

Definition

EXAMPLE 10-15 Consider the process manufacturing crankshaft bearings described in Example 8-6.

Suppose that a modification is made in the surface finishing process and that, subse-

quently, a second random sample of 85 axle shafts is obtained. The number of defective

shafts in this second sample is 8. Therefore, since n 1  85, n 2  85, and

pˆ 2  (^8)  85 0.09, we can obtain an approximate 95% confidence interval on the
pˆ 1 0.12,
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