Chapter 2 Simple Comparative Experiments Solutions

Solutions from Montgomery, D. C. (2001) Design and Analysis of Experiments, Wiley, NY

Chapter 2

Simple Comparative Experiments

Solutions

2-1 The breaking strength of a fiber is required to be at least 150 psi. Past experience has indicated that the standard deviation of breaking strength is V = 3 psi. A random sample of four specimens is tested. The results are y1=145, y2=153, y3=150 and y4=147.

(a) State the hypotheses that you think should be tested in this experiment.

H0: P = 150

H1: P > 150

(b) Test these hypotheses using D = 0.05. What are your conclusions?

n = 4, V = 3, y = 1/4 (145 + 153 + 150 + 147) = 148.75

zo

y Po V

148.75 150 3

1.25 3

0.8333

n

4

2

Since z0.05 = 1.645, do not reject. (c) Find the P-value for the test in part (b).

From the z-table: P # 1 >0.7967 2 30.7995 0.7967@ 0.2014

(d) Construct a 95 percent confidence interval on the mean breaking strength. The 95% confidence interval is

y zD 2

V n

d

P

d

y

zD 2

V n

148.75 1.963 2 d P d 148.75 1.963 2

145.81 d P d 151.69

2-2 The viscosity of a liquid detergent is supposed to average 800 centistokes at 25qC. A random sample of 16 batches of detergent is collected, and the average viscosity is 812. Suppose we know that the standard deviation of viscosity is V = 25 centistokes.

(a) State the hypotheses that should be tested.

H0: P = 800

H1: P z 800

(b) Test these hypotheses using D = 0.05. What are your conclusions?

2-1

Solutions from Montgomery, D. C. (2001) Design and Analysis of Experiments, Wiley, NY

zo

y Po V

812 800 25

12 25

1.92

n

16

4

Since zD/2 = z0.025 = 1.96, do not reject.

(c) What is the P-value for the test? P 2(0.0274) 0.0549

(d) Find a 95 percent confidence interval on the mean.

The 95% confidence interval is

y zD 2

V n

d P d y zD 2

V n

812 1.9625 4 d P d 812 1.9625 4

812 12.25 d P d 812 12.25

799.75 d P d 824.25

2-3 The diameters of steel shafts produced by a certain manufacturing process should have a mean diameter of 0.255 inches. The diameter is known to have a standard deviation of V = 0.0001 inch. A random sample of 10 shafts has an average diameter of 0.2545 inches.

(a) Set up the appropriate hypotheses on the mean P.

H0: P = 0.255 H1: P z 0.255 (b) Test these hypotheses using D = 0.05. What are your conclusions?

n = 10, V = 0.0001, y = 0.2545

Since z0.025 = 1.96, reject H0.

zo

y Po V

0.2545 0.255 0.0001

15.81

n

10

(c) Find the P-value for this test. P=2.6547x10-56

(d) Construct a 95 percent confidence interval on the mean shaft diameter.

The 95% confidence interval is

y zD 2

V n

d P d y zD 2

V n

0.2545

1.96

? ??

0.0001 10

? ??

d

P

d

0.2545

1.96

? ??

0.0001 10

? ??

0.254438 d P d 0.254562

2-4 A normally distributed random variable has an unknown mean P and a known variance V2 = 9. Find the sample size required to construct a 95 percent confidence interval on the mean, that has total width of 1.0.

2-2

Solutions from Montgomery, D. C. (2001) Design and Analysis of Experiments, Wiley, NY

Since y a N(P,9), a 95% two-sided confidence interval on P is

y zD2

V n

d

P

d

y zD2

V n

y (1.96) 3 d P d y (1.96) 3

n

n

If the total interval is to have width 1.0, then the half-interval is 0.5. Since z /2 = z0.025 = 1.96,

 1.96 3 n 0.5

n 1.963 0.5 11.76 n 11.762 138.30 # 139

2-5 The shelf life of a carbonated beverage is of interest. Ten bottles are randomly selected and tested, and the following results are obtained:

Days

108

138

124

163

124

159

106

134

115

139

(a) We would like to demonstrate that the mean shelf life exceeds 120 days. Set up appropriate hypotheses for investigating this claim.

H0: P = 120

H1: P > 120

(b) Test these hypotheses using D = 0.01. What are your conclusions?

y = 131

s2 = [ (108 - 131)2 + (124 - 131)2 + (124 - 131)2 + (106 - 131)2 + (115 - 131)2 + (138 - 131)2 + (163 - 131)2 + (159 - 131)2 + (134 - 131)2 + ( 139 - 131)2 ] / (10 - 1)

s2 = 3438 / 9 = 382 s 382 19.54

to

y Po sn

131 120 1.78 19.54 10

since t0.01,9 = 2.821; do not reject H0

Minitab Output T-Test of the Mean

Test of mu = 120.00 vs mu > 120.00

Variable

N

Shelf Life 10

Mean 131.00

StDev SE Mean

19.54

6.18

T Confidence Intervals

T 1.78

P 0.054

2-3

Solutions from Montgomery, D. C. (2001) Design and Analysis of Experiments, Wiley, NY

Variable

N

Shelf Life 10

Mean 131.00

StDev SE Mean

99.0 % CI

19.54

6.18 ( 110.91, 151.09)

(c) Find the P-value for the test in part (b). P=0.054

(d) Construct a 99 percent confidence interval on the mean shelf life.

The 95% confidence interval is y tD 2,n1

s n

dP

d

y

tD 2 ,n1

s n

131

3.250

? ??

1954 10

? ??

d

P

d

131

3.250

? ??

1954 10

? ??

110.91 d P d 151.09

2-6 Consider the shelf life data in Problem 2-5. Can shelf life be described or modeled adequately by a normal distribution? What effect would violation of this assumption have on the test procedure you used in solving Problem 2-5?

A normal probability plot, obtained from Minitab, is shown. There is no reason to doubt the adequacy of the normality assumption. If shelf life is not normally distributed, then the impact of this on the t-test in problem 2-5 is not too serious unless the departure from normality is severe.

Normal Probability Plot for Shelf Life ML Estimates

Percent

99

ML Estimates

Mean 131

95

StDev 18.5418

90

80

Goodness of Fit

70

AD*

1.292

60

50

40

30

20

10 5

1

86

96

106

116

126

136

146

156

166

176

Data

2-7 The time to repair an electronic instrument is a normally distributed random variable measured in hours. The repair time for 16 such instruments chosen at random are as follows:

Hours

159

280

101

212

224

379

179

264

222

362

168

250

149

260

485

170

2-4

Solutions from Montgomery, D. C. (2001) Design and Analysis of Experiments, Wiley, NY

(a) You wish to know if the mean repair time exceeds 225 hours. Set up appropriate hypotheses for investigating this issue. H0: P = 225 H1: P > 225

(b) Test the hypotheses you formulated in part (a). What are your conclusions? Use D = 0.05.

y = 247.50 s2 =146202 / (16 - 1) = 9746.80

s 9746.8 98.73

to

y Po s

241.50 225 98.73

0.67

n

16

since t0.05,15 = 1.753; do not reject H0

Minitab Output T-Test of the Mean

Test of mu = 225.0 vs mu > 225.0

Variable

N

Hours

16

Mean 241.5

StDev SE Mean

98.7

24.7

T Confidence Intervals

Variable

N

Hours

16

Mean 241.5

StDev SE Mean

98.7

24.7 (

T 0.67

P 0.26

95.0 % CI 188.9, 294.1)

(c) Find the P-value for this test. P=0.26

(d) Construct a 95 percent confidence interval on mean repair time.

The 95% confidence interval is y tD 2,n1

s n

dP

d

y tD 2 ,n1

s n

241.50

2.131

? ??

98.73 16

? ??

d

P

d

241.50

2.131

? ??

98.73 16

? ??

188.9 d P d 294.1

2-8 Reconsider the repair time data in Problem 2-7. Can repair time, in your opinion, be adequately modeled by a normal distribution?

The normal probability plot below does not reveal any serious problem with the normality assumption.

2-5

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