Applied Statistics and Probability for Engineers

difference in the proportion of defective bearings produced under the two processes from

Equation 10-39 as follows:

or

This simplifies to

This confidence interval includes zero, so, based on the sample data, it seems unlikely that the

changes made in the surface finish process have reduced the proportion of defective crank-

shaft bearings being produced.

EXERCISES FOR SECTION 10-6

0.06p 1 p 2 0.12

p 1 p 2 0.120.09 1.96

B

0.12 1 0.88 2

85

0.09 1 0.91 2

85

0.120.091.96

B

0.12 1 0.88 2

85

0.09 1 0.91 2

85

p 1 p 2 pˆ 1 pˆ 2 z0.025

B

pˆ 111 pˆ 12

n 1

pˆ 211 pˆ 22

n 2

pˆ 1 pˆ 2 z0.025

B

pˆ 111 pˆ 12

n 1

pˆ 211 pˆ 22

n 2

366 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES

10-61. Two different types of injection-molding machines

are used to form plastic parts. A part is considered defective

if it has excessive shrinkage or is discolored. Two random

samples, each of size 300, are selected, and 15 defective

parts are found in the sample from machine 1 while 8 defec-

tive parts are found in the sample from machine 2. Is it rea-

sonable to conclude that both machines produce the same

fraction of defective parts, using  0.05? Find the P-value

for this test.

10-62. Two different types of polishing solution are being

evaluated for possible use in a tumble-polish operation for

manufacturing interocular lenses used in the human eye fol-

lowing cataract surgery. Three hundred lenses were tumble-

polished using the first polishing solution, and of this number

253 had no polishing-induced defects. Another 300 lenses

were tumble-polished using the second polishing solution, and

196 lenses were satisfactory upon completion. Is there any

reason to believe that the two polishing solutions differ? Use

 0.01. Discuss how this question could be answered with a

confidence interval on p 1  p 2.

10-63. Consider the situation described in Exercise 10-61.

Suppose that p 1  0.05 and p 2  0.01.

(a) With the sample sizes given here, what is the power of the

test for this two-sided alternate?

(b) Determine the sample size needed to detect this difference

with a probability of at least 0.9. Use  0.05.

10-64. Consider the situation described in Exercise 10-61.

Suppose that p 1  0.05 and p 2  0.02.

(a) With the sample sizes given here, what is the power of the

test for this two-sided alternate?

(b) Determine the sample size needed to detect this difference

with a probability of at least 0.9. Use  0.05.

10-65. A random sample of 500 adult residents of Maricopa

County found that 385 were in favor of increasing the high-

way speed limit to 75 mph, while another sample of 400 adult

residents of Pima County found that 267 were in favor of the

increased speed limit. Do these data indicate that there is a dif-

ference in the support for increasing the speed limit between

the residents of the two counties? Use  0.05. What is the

P-value for this test?

10-66. Construct a 95% confidence interval on the differ-

ence in the two fractions defective for Exercise 10-61.

10-67. Construct a 95% confidence interval on the differ-

ence in the two proportions for Exercise 10-65. Provide a

practical interpretation of this interval.

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