Chapter 2 Simple Comparative Experiments PDF

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Solutions from Montgomery, D. C. (2004) 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 σ = 3 ps...

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Solutions from Montgomery, D. C. (2004) 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 σ = 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: µ = 150

H1: µ > 150

(b) Test these hypotheses using α = 0.05. What are your conclusions? n = 4, σ = 3, y = 1/4 (145 + 153 + 150 + 147) = 148.75 zo =

y − µo

σ

n

=

148.75 − 150 −1.25 = = −0.8333 3 3 2 4

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 − zα 2

σ

≤ µ ≤ y + zα 2

σ

n n 148.75 − (1.96 )(3 2) ≤ µ ≤ 148.75 + (1.96 )(3 2)

145. 81 ≤ µ ≤ 151. 69

2-2 The viscosity of a liquid detergent is supposed to average 800 centistokes at 25°C. 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 σ = 25 centistokes. (a) State the hypotheses that should be tested. H0: µ = 800

H1: µ ≠ 800

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

2-1

Solutions from Montgomery, D. C. (2004) Design and Analysis of Experiments, Wiley, NY zo =

y − µo

=

σ

n

812 − 800 12 = = 1.92 25 25 4 16

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

P = 2(0.0274) = 0.0549

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

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

σ

σ

The 95% confidence interval is

y − zα 2

≤ µ ≤ y + zα 2

812 − (1.96 )(25 4) ≤ µ ≤ 812 + (1.96 )(25 4 ) 812 − 12.25 ≤ µ ≤ 812 + 12.25 799.75 ≤ µ ≤ 824.25

n

n

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 σ = 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 µ. H0: µ = 0.255

H1: µ ≠ 0.255

(b) Test these hypotheses using α = 0.05. What are your conclusions? n = 10, σ = 0.0001, y = 0.2545

zo =

y − µo

σ

=

n

0.2545 − 0.255 = −15.81 0.0001 10

Since z0.025 = 1.96, reject H0. (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 − zα 2

σ n

≤ µ ≤ y + zα 2

σ n

⎛ 0.0001 ⎞ ⎛ 0.0001 ⎞ 0.2545 − (1.96 ) ⎜ ⎟ ≤ µ ≤ 0.2545 + (1.96 ) ⎜ ⎟ ⎝ 10 ⎠ ⎝ 10 ⎠

0. 254438 ≤ µ ≤ 0. 254562

2-4 A normally distributed random variable has an unknown mean µ and a known variance σ2 = 9. Find the sample size required to construct a 95 percent confidence interval on the mean, that has total length of 1.0.

2-2

Solutions from Montgomery, D. C. (2004) Design and Analysis of Experiments, Wiley, NY Since y ∼ N(µ,9), a 95% two-sided confidence interval on µ is

σ

y − zα 2

y − (196 . )

n

≤ µ ≤ y + zα 2 3 n

σ

n

≤ µ ≤ y + (196 . )

3 n

If the total interval is to have width 1.0, then the half-interval is 0.5. Since zα/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 108 124 124 106 115

138 163 159 134 139

(a) We would like to demonstrate that the mean shelf life exceeds 120 days. Set up appropriate hypotheses for investigating this claim. H0: µ = 120

H1: µ > 120

(b) Test these hypotheses using α = 0.01. What are your conclusions? y = 131 S2 = 3438 / 9 = 382 S = 382 = 19.54

to =

y − µo S

n

=

131 − 120 19.54

10

= 1.78

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 Shelf Life

N 10

Mean 131.00

StDev 19.54

Mean 131.00

StDev 19.54

SE Mean 6.18

T 1.78

P 0.054

T Confidence Intervals Variable Shelf Life

N 10

SE Mean 6.18

(

2-3

99.0 % CI 110.91, 151.09)

Solutions from Montgomery, D. C. (2004) Design and Analysis of Experiments, Wiley, NY

(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. S S ≤ µ ≤ y + tα 2,n−1 with α = 0.01. The 99% confidence interval is y − tα 2,n−1 n n ⎛ 1954 ⎞ ⎛ 1954 ⎞ 131 − ( 3.250 ) ⎜ ⎟ ≤ µ ≤ 131 + ( 3.250 ) ⎜ ⎟ ⎝ 10 ⎠ ⎝ 10 ⎠

110.91 ≤ µ ≤ 15109 .

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

99

ML Estimates

95

Mean

131

StDev

18.5418

90

Goodness of Fit

Percent

80

AD*

70 60 50 40 30

1.292

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 159 224 222 149

280 379 362 260

101 179 168 485

212 264 250 170

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

2-4

Solutions from Montgomery, D. C. (2004) Design and Analysis of Experiments, Wiley, NY H0: µ = 225

H1: µ > 225

(b) Test the hypotheses you formulated in part (a). What are your conclusions? Use α = 0.05. y = 247.50 S2 =146202 / (16 - 1) = 9746.80 S = 9746.8 = 98.73

to =

y − µo 241.50 − 225 = = 0.67 98.73 S 16 n

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 Hours

N 16

Mean 241.5

StDev 98.7

Mean 241.5

StDev 98.7

SE Mean 24.7

T 0.67

P 0.26

T Confidence Intervals Variable Hours

N 16

SE Mean 24.7

(

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 − tα 2,n−1

S n

≤ µ ≤ y + tα 2,n−1

S n

⎛ 98.73 ⎞ ⎛ 98.73 ⎞ 241.50 − ( 2.131) ⎜ ⎟ ≤ µ ≤ 241.50 + ( 2.131) ⎜ ⎟ 16 ⎝ ⎠ ⎝ 16 ⎠

188.9 ≤ µ ≤ 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

Solutions from Montgomery, D. C. (2004) Design and Analysis of Experiments, Wiley, NY Normal Probability Plot for Hours ML Estimates

99

ML Estimates

95

Mean

241.5

StDev

95.5909

90

Goodness of Fit

Percent

80

AD*

70 60 50 40 30

1.185

20 10 5

1

50

150

250

350

450

Data

2-9 Two machines are used for filling plastic bottles with a net volume of 16.0 ounces. The filling processes can be assumed to be normal, with standard deviation of σ1 = 0.015 and σ2 = 0.018. The quality engineering department suspects that both machines fill to the same net volume, whether or not this volume is 16.0 ounces. An experiment is performed by taking a random sample from the output of each machine. Machine 1 16.03 16.01 16.04 15.96 16.05 15.98 16.05 16.02 16.02 15.99

Machine 2 16.02 16.03 15.97 16.04 15.96 16.02 16.01 16.01 15.99 16.00

(a) State the hypotheses that should be tested in this experiment. H0: µ1 = µ2

H1: µ1 ≠ µ2

(b) Test these hypotheses using α=0.05. What are your conclusions? y1 = 16. 015 σ 1 = 0. 015 n1 = 10 zo =

y2 = 16. 005 σ 2 = 0. 018 n2 = 10

y1 − y2

σ 12 n1

+

σ 22 n2

=

16. 015 − 16. 018

0. 0152 0. 0182 + 10 10

= 1. 35

z0.025 = 1.96; do not reject (c) What is the P-value for the test? P = 0.1770 (d) Find a 95 percent confidence interval on the difference in the mean fill volume for the two machines.

2-6

Solutions from Montgomery, D. C. (2004) Design and Analysis of Experiments, Wiley, NY The 95% confidence interval is

σ 12

y1 − y 2 − z α 2

(16.015 − 16.005) − (19.6)

+

n1

σ 22 n2

≤ µ 1 − µ 2 ≤ y1 − y 2 + z α 2

σ 12 n1

+

σ 22 n2

0.015 0.018 0.015 2 0.018 2 + ≤ µ1 − µ 2 ≤ (16.015 − 16.005) + (19.6) + 10 10 10 10 − 0.0045 ≤ µ 1 − µ 2 ≤ 0.0245 2

2

2-10 Two types of plastic are suitable for use by an electronic calculator manufacturer. The breaking strength of this plastic is important. It is known that σ1 = σ2 = 1.0 psi. From random samples of n1 = 10 and n2 = 12 we obtain y 1 = 162.5 and y 2 = 155.0. The company will not adopt plastic 1 unless its breaking strength exceeds that of plastic 2 by at least 10 psi. Based on the sample information, should they use plastic 1? In answering this questions, set up and test appropriate hypotheses using α = 0.01. Construct a 99 percent confidence interval on the true mean difference in breaking strength. H0: µ1 - µ2 =10

H1: µ1 - µ2 >10

y1 = 162.5

y2 = 155.0

σ1 = 1

σ2 = 1

n1 = 10

zo =

n2 = 10

y1 − y2 − 10

σ 12 n1

+

σ 22

=

162. 5 − 155. 0 − 10 12 12 + 10 12

n2

= −5.85

z0.01 = 2.225; do not reject The 99 percent confidence interval is

y1 − y 2 − z α 2

(162.5 − 155.0) − (2.575)

σ 12 n1

+

σ 22 n2

≤ µ 1 − µ 2 ≤ y1 − y 2 + z α 2

σ 12 n1

+

σ 22 n2

12 12 12 12 + ≤ µ1 − µ 2 ≤ (162.5 − 155.0) + (2.575) + 10 12 10 12 6.40 ≤ µ 1 − µ 2 ≤ 8.60

2-11 The following are the burning times (in minutes) of chemical flares of two different formulations. The design engineers are interested in both the means and variance of the burning times.

Type 1 65 81 57 66 82

Type 2 64 71 83 59 65

82 67 59 75 70

56 69 74 82 79

(a) Test the hypotheses that the two variances are equal. Use α = 0.05.

2-7

Solutions from Montgomery, D. C. (2004) Design and Analysis of Experiments, Wiley, NY

H 0 : σ 12 = σ 22 H1 : σ ≠ σ 2 1

F0 =

F0.025,9 ,9 = 4.03

2 2 S12 S 22

F0.975,9 ,9 =

S1 = 9.264 S 2 = 9.367 =

8582 . = 0.98 87.73

1 1 = = 0.248 Do not reject. F0.025,9 ,9 4.03

(b) Using the results of (a), test the hypotheses that the mean burning times are equal. Use α = 0.05. What is the P-value for this test? S p2 =

(n1 − 1) S12 + (n 2 − 1) S 22 156195 . = = 86.775 n1 + n 2 − 2 18

S p = 9.32

y1 − y 2

t0 =

Sp

1 1 + n1 n 2

=

70.4 − 70.2

1 1 + 9.32 10 10

= 0.048

t 0.025,18 = 2.101 Do not reject.

From the computer output, t=0.05; do not reject. Also from the computer output P=0.96 Minitab Output Two Sample T-Test and Confidence Interval Two sample T for Type 1 vs Type 2

Type 1 Type 2

N 10 10

Mean 70.40 70.20

StDev 9.26 9.37

SE Mean 2.9 3.0

95% CI for mu Type 1 - mu Type 2: ( -8.6, 9.0) T-Test mu Type 1 = mu Type 2 (vs not =): T = 0.05 Both use Pooled StDev = 9.32

P = 0.96

DF = 18

(c) Discuss the role of the normality assumption in this problem. Check the assumption of normality for both types of flares. The assumption of normality is required in the theoretical development of the t-test. However, moderate departure from normality has little impact on the performance of the t-test. The normality assumption is more important for the test on the equality of the two variances. An indication of nonnormality would be of concern here. The normal probability plots shown below indicate that burning time for both formulations follow the normal distribution.

2-8

Solutions from Montgomery, D. C. (2004) Design and Analysis of Experiments, Wiley, NY Normal Probability Plot for Type 1 ML Estimates

99

ML Estimates

95

Mean

70.4

StDev

8.78863

90

Goodness of Fit

Percent

80

AD*

70 60 50 40 30

1.387

20 10 5

1

50

60

70

80

90

Data Normal Probability Plot for Type 2 ML Estimates

99

ML Estimates

95

Mean

70.2

StDev

8.88594

90

Goodness of Fit

Percent

80

AD*

70 60 50 40 30

1.227

20 10 5

1

50

60

70

80

90

Data

2-12 An article in Solid State Technology, "Orthogonal Design of Process Optimization and Its Application to Plasma Etching" by G.Z. Yin and D.W. Jillie (May, 1987) describes an experiment to determine the effect of C2F6 flow rate on the uniformity of the etch on a silicon wafer used in integrated circuit manufacturing. Data for two flow rates are as follows:

C2F6 (SCCM) 125 200

1 2.7 4.6

2 4.6 3.4

Uniformity Observation 3 4 5 2.6 3.0 3.2 2.9 3.5 4.1

(a) Does the C2F6 flow rate affect average etch uniformity? Use α = 0.05. No, C2F6 flow rate does not affect average etch uniformity.

2-9

6 3.8 5.1

Solutions from Montgomery, D. C. (2004) Design and Analysis of Experiments, Wiley, NY

Minitab Output Two Sample T-Test and Confidence Interval Two sample T for Uniformity Flow Rat 125 200

N 6 6

Mean 3.317 3.933

StDev 0.760 0.821

SE Mean 0.31 0.34

95% CI for mu (125) - mu (200): ( -1.63, 0.40) T-Test mu (125) = mu (200) (vs not =): T = -1.35 Both use Pooled StDev = 0.791

P = 0.21

DF = 10

(b) What is the P-value for the test in part (a)? From the computer printout, P=0.21 (c) Does the C2F6 flow rate affect the wafer-to-wafer variability in etch uniformity? Use α = 0.05.

H 0 : σ 12 = σ 22 H1 : σ 12 ≠ σ 22

F0.05,5,5 = 5.05 F0 =

0.5776 = 0.86 0.6724

Do not reject; C2F6 flow rate does not affect wafer-to-wafer variability. (d) Draw box plots to assist in the interpretation of the data from this experiment. The box plots shown below indicate that there is little difference in uniformity at the two gas flow rates. Any observed difference is not statistically significant. See the t-test in part (a).

Uniformity

5

4

3

125

200

Flow Rate

2-13 A new filtering device is installed in a chemical unit. Before its installation, a random sample 2 yielded the following information about the percentage of impurity: y 1 = 12.5, S1 =101.17, and n1 = 8. 2

After installation, a random sample yielded y 2 = 10.2, S2 = 94.73, n2 = 9.

(a) Can you concluded that the two variances are equal? Use α = 0.05.

2-10

Solutions from Montgomery, D. C. (2004) Design and Analysis of Experiments, Wiley, NY H 0 : σ 12 = σ 22 H1 : σ 12 ≠ σ 22

F0.025 ,7 ,8 = 4.53 F0 =

S12 S 22

=

101.17 = 1.07 94.73

Do Not Reject. Assume that the variances are equal. (b) Has the filtering device reduced the percentage of impurity significantly? Use α = 0.05. H 0 : µ1 = µ 2 H1 : µ1 ≠ µ 2

S p2 =

( n1 − 1 )S12 + ( n2 − 1 )S 22 ( 8 − 1 )( 101.17 ) + ( 9 − 1 )( 94.73 ) = = 97.74 n1 + n2 − 2 8+9−2

S p = 9.89

y1 − y2

t0 =

Sp

1 1 + n1 n2

t0.05 ,15 = 1.753

=

12.5 − 10.2

1 1 + 9.89 8 9

= 0.479

Do not reject. There is no evidence to indicate that the new filtering device has affected the mean

2-14 Photoresist is a light-sensitive material applied to semiconductor wafers so that the circuit pattern can be imaged on to the wafer. After application, the coated wafers are baked to remove the solvent in the photoresist mixture and to harden the resist. Here are measurements of photoresist thickness (in kÅ) for eight wafers baked at two different temperatures. Assume that all of the runs were made in random order.

95 ºC 11.176 7.089 8.097 11.739 11.291 10.759 6.467 8.315

100 ºC 5.263 6.748 7.461 7.015 8.133 7.418 3.772 8.963

(a) Is there evidence to support the claim that the higher baking temperature results in wafers with a lower mean photoresist thickness? Use α = 0.05.

2-11

Solutions from Montgomery, D. C. (2004) Design and Analysis of Experiments, Wiley, NY H 0 : µ1 = µ2 H1 : µ1 ≠ µ2

S p2 =

(n1 − 1) S12 + (n2 − 1) S22 (8 − 1)(4.41) + (8 − 1)(2.54) = 3.48 = 8+8−2 n1 + n2 − 2

S p = 1.86

y1 − y2

t0 =

Sp

1 1...