Applied Statistics and Probability for Engineers,7th edition 2017

(c) Analysis of Variance for LEAKAGE VOLTAGE

Source DF SS MS F P

LENGTH 3 8.1775 2.7258 6.16 0.009

13.4.8 An article in the Food Technology Journal (1956, Vol. 10, pp. 39–42) described a study on the protopectin content of

tomatoes during storage. Four storage times were selected, and samples from nine lots of tomatoes were analyzed. The

protopectin content (expressed as hydrochloric acid soluble fraction mg/kg) is in Table 13E-1.

(a) The researchers in this study hypothesized that mean protopectin content would be different at different storage

times. Can you confirm this hypothesis with a statistical test using

= 0.05?

(b) Find the P-value for the test in part (a).

(c) Which specific storage times are different? Would you agree with the statement that protopectin content decreases

as storage time increases?

(d) Analyze the residuals from this experiment.

TABLE 13E-1 Protopectin Content of Tomatoes in Storage

Storage

Time

Lot

1

2

3

4

5

6

7

8

9

0 days

1694.0

989.0

917.3

346.1

1260.0

965.6

1123.0

1106.0

1116.0

7 days

1802.0

1074.0

278.8

1375.0

544.0

672.2

818.0

406.8

461.6

14 days

1568.0

646.2

1820.0

1150.0

983.7

395.3

422.3

420.0

409.5

21 days

415.5

845.4

377.6

279.4

447.8

272.1

394.1

356.4

351.2

(a) Analysis of Variance of Propectin

Source DF SS MS F P

(c)

Fisher's pairwise comparisons

Family error rate = 0.196

13-37

Applied Statistics and Probability for Engineers,7th edition 2017

13-38

(d) Observations from lot 3 at 14 days appear unusual. Otherwise, the residuals are acceptable.

Supplemental Exercises

13.S9 Consider the following computer output.

Source

DF

SS

MS

F

P

Factor

?

126.880

63.4401

?

?

Block

?

54.825

18.2751

Error

6

?

2.7403

Total

11

198.147

(a) How many levels of the factor were used in this experiment?

(b) How many blocks were used?

(c) Fill in the missing information. Use bounds for the P-value.

(d) What conclusions would you draw if

= 0.05? What if

= 0.01?

Applied Statistics and Probability for Engineers,7th edition 2017

13-39

13.S10 An article in Lubrication Engineering (December 1990) described the results of an experiment designed to investigate

the effects of carbon material properties on the progression of blisters on carbon face seals. The carbon face seals are

used extensively in equipment such as air turbine starters. Five different carbon materials were tested, and the surface

roughness was measured. The data are as follows:

Carbon

Material Type

Surface Roughness

EC10

0.50

0.55

0.55

0.36

EC10A

0.31

0.07

0.25

0.18

0.56

0.20

EC4

0.20

0.28

0.12

EC1

0.10

0.16

(a) Does carbon material type have an effect on mean surface roughness? Use

= 0.05.

(b) Find the residuals for this experiment. Does a normal probability plot of the residuals indicate any problem with the

normality assumption?

(c) Plot the residuals versus

ˆij

y

. Comment on the plot.

(d) Find a 95% confidence interval on the difference between mean surface roughness for the EC10 and the EC1

carbon grades.

(e) Apply the Fisher LSD method to this experiment. Summarize your conclusions regarding the effect of material type

on surface roughness.

(a)

Analysis of Variance for SURFACE ROUGNESS

Source DF SS MS F P

Material 3 0.2402 0.0801 4.96 0.020

Applied Statistics and Probability for Engineers,7th edition 2017

13-40

(b) One observation is an outlier.

(c) There appears to be a problem with constant variance. This may be due to the outlier in the data.

Applied Statistics and Probability for Engineers,7th edition 2017

13-41

(d) 95% Confidence interval on the difference in the means of EC10 and EC1

13.S11 An article in the IEEE Transactions on Components, Hybrids, and Manufacturing Technology [(1992, Vol. 15(2), pp.

146–153)] described an experiment in which the contact resistance of a brake-only relay was studied for three different

materials (all were silver-based alloys). The data are as follows.

Alloy

Contact Resistance

1

95

97

99

98

99

99

99

94

95

98

2

104

102

102

105

99

102

111

103

100

103

3

119

130

132

136

141

172

145

150

144

135

(a) Does the type of alloy affect mean contact resistance? Use

= 0.01.

(b) Use Fisher’s LSD method to determine which means differ.

(c) Find a 99% confidence interval on the mean contact resistance for alloy 3.

(d) Analyze the residuals for this experiment.

(a) Analysis of Variance for RESISTANCE

Source DF SS MS F P

ALLOY 2 10941.8 5470.9 76.09 0.000

(b) Fisher's pairwise comparisons

Family error rate = 0.119

(c) 99% Confidence interval on the mean contact resistance for alloy 3

3 0.005,27 3 0.005,27

EE

i

M S M S

y t y t

nn

− +

Applied Statistics and Probability for Engineers,7th edition 2017

13-42

(d) Variability of the residuals increases with the response. The normal probability plot has some curvature in the tails,

indicating a problem with the normality assumption. A transformation of the response should be conducted.

13.S12 An article in the Journal of Quality Technology [(1982, Vol. 14(2), pp. 80–89)] described an experiment in which three

different methods of preparing fish were evaluated on the basis of sensory criteria, and a quality score was assigned.

Assume that these methods have been randomly selected from a large population of preparation methods. The data are

in the following table:

Method

Score

1

24.4

23.2

25.0

19.7

22.2

24.4

23.8

18.0

2

22.1

19.5

17.3

19.7

22.3

23.2

21.4

22.6

3

23.3

22.8

22.4

23.7

20.4

23.5

20.8

24.1

(a) Is there any difference in preparation methods? Use

= 0.05.

(b) Calculate the P-value for the F-statistic in part (a).

(c) Analyze the residuals from this experiment and comment on model adequacy.

(d) Estimate the components of variance.

(a) Analysis of Variance for SCORE

Source DF SS MS F P

Applied Statistics and Probability for Engineers,7th edition 2017

13-43

13.S13 An article in the Journal of Agricultural Engineering Research (1992, Vol. 52, pp. 53–76) described an experiment to

investigate the effect of drying temperature of wheat grain on baking quality bread. Three temperature levels were

used, and the response variable measured was the volume of the loaf of bread produced. The data are as follows:

Temperature (°C)

Volume (CC)

70.0

1245

1235

1285

1245

1235

75.0

1235

1240

1200

1220

1210

80.0

1225

1200

1170

1155

1095

(a) Does drying temperature affect mean bread volume? Use

= 0.01.

(b) Find the P-value for this test.

(c) Use the Fisher LSD method to determine which means are different.

(d) Analyze the residuals from this experiment and comment on model adequacy.

(a) Analysis of Variance for VOLUME

Source DF SS MS F P

Applied Statistics and Probability for Engineers,7th edition 2017

13-44

(d) There are some relatively small differences in the variability at the different levels of temperature. The variability

decreases with the fitted values. There is an unusual observation on the normal probability plot.

13.S14 An article in Nature Genetics [(2003, Vol. 34(1), pp. 85–90)], “Treatment-Specific Changes in Gene Expression

Discriminate in vivo Drug Response in Human Leukemia Cells,” reported the results of a study of gene expression as a

function of different treatments for leukemia. Three treatment groups are mercaptopurine (MP) only, low-dose

methotrexate (LDMTX) and MP, and high-dose methotrexate (HDMTX) and MP. Each group contained 10 subjects.

The responses from a specific gene are shown in Table 13E-3.

(a) Check the normality of the data. Can you assume that these samples are from normal populations?

(b) Take the logarithm of the raw data and check the normality of the transformed data. Is there evidence to support the

claim that the treatment means differ for the transformed data? Use

= 0.1.

(c) Analyze the residuals from the transformed data and comment on model adequacy.

Applied Statistics and Probability for Engineers,7th edition 2017

13-45

TABLE 13E-3 Treatment-Specific Changes in Gene Expression

Treatments

Observations

MP ONLY

334.5

31.6

701

41.2

61.2

69.6

67.5

66.6

120.7

881.9

MP + HDMTX

919.4

404.2

1024.8

54.1

62.8

671.6

882.1

354.2

321.9

91.1

MP + LDMTX

108.4

26.1

240.8

191.1

69.7

242.8

62.7

396.9

23.6

290.4

(a) The normal probability plot shows that the normality assumption is not reasonable.

(b) The normal probability plot shows that the normality assumption is reasonable.

Applied Statistics and Probability for Engineers,7th edition 2017

13-46

(c) The normal probability plot and the residual plots show that the model assumptions are reasonable.

13.S15 An article in Communications of the ACM [(1987, Vol. 30(5), pp. 53–76] reported on a study of different algorithms for

estimating software development costs. Six algorithms were applied to eight software development projects and the

percent error in estimating the development cost was observed. The data are in Table 13E-2.

(a) Do the algorithms differ in mean cost estimation accuracy? Use

= 0.05.

(b) Analyze the residuals from this experiment.

(c) Which algorithm would you recommend for use in practice?

TABLE 13E-2 Software Development Costs

Project

Algorithm

1

2

3

4

5

6

7

8

1 (SLIM)

1244

21

82

2221

905

839

527

122

2(COCOMO-A)

281

129

396

1306

336

910

473

199

3(COCOMO-R)

220

84

458

543

300

794

488

142

4(COCOMO-C)

225

83

425

552

291

826

509

153

5(FUNCTION POINTS)

19

11

−34

121

15

103

87

−17

6(ESTIMALS)

−20

35

−53

170

104

199

142

41

Applied Statistics and Probability for Engineers,7th edition 2017

(a) Analysis of Variance for PCTERROR

Source DF SS MS F P

(b) The residuals look acceptable, except there is one unusual point.

13-48

13.S16 An article in Journal of Hazardous Materials [“Toxicity Assessment from Electro-Coagulation Treated-Textile Dye

Waste Waters by Bioassays,” 2009, Vol. 172(1), pp. 330–337] discussed a study of pollutant removal from textile

dyeing waste water with an electro-coagulation technique. Chemical oxygen demand (COD) (a common measure of

water pollution) was used as the response, and three different values for electrolysis time were considered. The

following data were extracted from a larger study. Suppose that a randomized complete block experiment was

conducted with three blocks based on initial pH values.

Electrolysis

time (min)

Initial pH

3

7

11

15

77.1

75.2

42.2

30

80.1

76.8

45.0

45

82.8

75.2

46.8

(a) Is there an effect of electrolysis time at

= 0.05? Calculate the P-value.

(b) Analyze the residuals from the experiment.

(c) Calculate a 95% confidence interval on mean COD removal when the electrolysis time is 15 minutes.

(d) Perform an ANOVA assuming that all data are collected at a single pH value. Comment on differences from

part (a).

(a)

Source DF SS MS F P

Time 2 18.81 9.40 3.80 0.119

Individual 95% CIs For Mean Based on Pooled StDev

Time Mean -+---------+---------+---------+--------

Applied Statistics and Probability for Engineers,7th edition 2017

13-49

(b)

(c) t0.05/2,4 = 2.7764, and the pooled standard deviation is 1.574.

Applied Statistics and Probability for Engineers,7th edition 2017

(d)

Source DF SS MS F P

Time 2 19 9 0.03 0.975

Individual 95% CIs For Mean Based on

Pooled StDev

Level N Mean StDev -----+---------+---------+---------+----

## Trusted by Thousands of

Students

Here are what students say about us.

###### Resources

###### Company

Copyright ©2022 All rights reserved. | CoursePaper is not sponsored or endorsed by any college or university.