Probability and Statistics for Engineering and the Sciences
9th Edition
ISBN: 9781305251809
Author: Jay L. Devore
Publisher: Cengage Learning
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Chapter 13.4, Problem 44E
The accompanying Minitab regression output is based on data that appeared in the article “Application of Design of Experiments for Modeling Surface Roughness in Ultrasonic Vibration Turning” (J. of Engr. Manuf., 2009: 641–652). The response variable is surface roughness (μm), and the independent variables are vibration amplitude (μm), depth of cut (μm), feed rate (mm/rev), and cutting speed (m/min), respectively.
- a. How many observations were there in the data set?
- b. Interpret the coefficient of multiple determination.
- c. Carry out a test of hypotheses to decide if the model specifies a useful relationship between the response variable and at least one of the predictors.
- d. Interpret the number 18.2602 that appears in the Coef column.
- e. At significance level .10, can any single one of the predictors be eliminated from the model provided that all of the other predictors are retained?
- f. The estimated SD of
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Wrinkle recovery angle and tensile strength are the two most important characteristics for evaluating the performance of crosslinked cotton fabric. An increase in the degree of crosslinking, as determined by ester carboxyl band absorbance, improves the wrinkle
resistance of the fabric (at the expense of reducing mechanical strength). The accompanying data on x = absorbance and y = wrinkle resistance angle was read from a graph in the paper "Predicting the Performance of Durable Press Finished Cotton Fabric with
Infrared Spectroscopy".†
x 0.115 0.126 0.183 0.246 0.282 0.344 0.355 0.452 0.491 0.554 0.651
y 334 342 355 363
365 372 381 392
400 412 420
Here is regression output from Minitab:
Predictor
Constant
absorb
S = 3.60498
Coef
321.878
156.711
SOURCE
Regression
Residual Error
Total
SE Coef
2.483
6.464
R-Sq = 98.5%
DF
1
9
10
SS
7639.0
117.0
7756.0
T
129.64
24.24
0.000
0.000
R-Sq (adj) = 98.3%
MS
7639.0
13.0
F
P
587.81
(a) Does the simple linear regression model appear to be…
Wrinkle recovery angle and tensile strength are the two most important characteristics for evaluating the performance of crosslinked cotton fabric. An increase in the degree of crosslinking, as determined by ester carboxyl band absorbance, improves the wrinkle resistance of the fabric (at the expense of reducing mechanical
strength). The accompanying data on x = absorbance and y = wrinkle resistance angle was read from a graph in the paper "Predicting the Performance of Durable Press Finished Cotton Fabric with Infrared Spectroscopy".t
半
0.115 0.126 0.183 0.246 0.282 0.344 0.355 0.452 0.491 0.554 0.651
334 342
355
363
365
372
381
392
400
412
420
Here is regression output from Minitab:
Predictor
Coef
SE Coef
P
Constant
321.878
2.483
129.64
0.000
absorb
156.711
6.464
24.24
0.000
S = 3.60498
R-Sq = 98.5%
R-Są (adj) - 98.3%
SOURCE
DF
MS
F
P
Regression
1
7639.0
7639.0
587.81
0.000
Residual Error
9
117.0
13.0
Total
10
7756.0
(a) Does the simple linear regression model appear to be…
The accompanying Minitab regression output is based on data that appeared in the article "Application of Design of Experiments for Modeling Surface Roughness in Ultrasonic Vibration Turning."+ The response variable is surface roughness (um), and the independent variables are vibration amplitude (um), depth of cut (mm), feed rate (mm/rev), and cutting speed (m/min), respectively.
The regression equation is
Ra = -0.972 - 0.0312a + 0.557d + 18.3f + 0.00282v
Predictor
Coef
SE Coef
Constant
-0.9723
0.3923
-2.48
0.015
-0.03117
0.01864
-1.67
0.099
d
0.5568
0.3185
1.75
0.084
18.2602
0.7536
24.23
0.000
0.002822
0.003977
0.71
0.480
S = 0.822059
Source
R-Sq = 88.6
R-Sq (adj) = 88.04
DS
MS
0.000
Regression
Residual Error
4
401.02
100.25
148.35
76
51.36
0.68
Total
80
452.38
(a) How many observations were there in the data set?
observations
(b) Interpret the coefficient of multiple determination.
O 8.0% of the observed variation in feed rate can be explained by the model relationship with vibration…
Chapter 13 Solutions
Probability and Statistics for Engineering and the Sciences
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