(a) What proportion of total variability is explained by this model?

(b) Are there any influential points in these data?

SOLUTION

The regression equation is

Rating Pts 2.99 1.20Pct Comp 4.60Pct TD 3.81Pct Int= + + −

Predictor

Coef

Constant

2.986

Pct Comp

1.19857

0.00706

0.01198

0.00413

0.00465

0.03510

0.00015

0.04000

0.00015

distance greater than 1, two points are different and might be further studied for influence.

Reserve Problems Chapter 12 Section 5 Problem 7

Heat treating is often used to carburize metal parts such as gears. The thickness of the carburized

layer is considered a crucial feature of the gear and contributes to the overall reliability of the

part. Because of the critical nature of this feature, two different lab tests are performed on each

furnace load. One test is run on a sample pin that accompanies each load. The other test is a

destructive test that cross-sections an actual part. This test involves running a carbon analysis on

the surface of both the gear pitch (top of the gear tooth) and the gear root (between the gear

teeth). Table given below shows the results of the pitch carbon analysis test for 32 parts.

The regressors are furnace temperature (TEMP), carbon concentration and duration of the

carburizing cycle (SOAKPCT, SOAKTIME), and carbon concentration and duration of the

diffuse cycle (DIFFPCT, DIFFTIME). The response is the result of the pitch carbon analysis test

(PITCH).

TEMP

SOAKTIME

SOAKPCT

DIFFTIME

DIFFPCT

PITCH

1650

0.58

1.10

0.25

0.90

0.013

1650

0.66

1.10

0.33

0.90

0.035

1650

0.66

1.10

0.33

0.90

0.015

1650

0.66

1.10

0.33

0.95

0.016

1600

0.66

1.15

0.33

1.00

0.015

1600

0.66

1.15

0.33

1.00

0.016

1650

1.00

1.10

0.50

0.80

0.033

1650

1.17

1.10

0.58

0.80

0.021

1650

1.17

1.10

0.58

0.80

0.018

1650

1.17

1.10

0.58

0.80

0.019

1650

1.17

1.10

0.58

0.90

0.040

1650

1.17

1.10

0.58

0.90

0.019

1650

1.17

1.15

0.58

0.90

0.021

1650

1.20

1.15

1.10

0.80

0.025

1650

2.00

1.15

1.00

0.80

0.025

1650

2.00

1.10

1.10

0.80

0.026

1650

2.20

1.10

1.10

0.80

0.024

1650

2.20

1.10

1.10

0.80

0.025

1650

2.20

1.15

1.10

0.80

0.024

1650

2.20

1.10

1.10

0.90

0.025

1650

2.20

1.10

1.10

0.90

0.027

1650

2.20

1.10

1.50

0.90

0.026

1650

3.00

1.15

1.50

0.80

0.029

1650

3.00

1.10

1.50

0.70

0.030

1650

3.00

1.10

1.50

0.75

0.028

1650

3.00

1.15

1.66

0.85

0.032

1650

3.33

1.10

1.50

0.80

0.033

1700

4.00

1.10

1.50

0.70

0.039

1650

4.00

1.10

1.50

0.70

0.040

1650

4.00

1.15

1.50

0.85

0.035

1700

12.50

1.00

1.50

0.70

0.056

1700

18.50

1.00

1.50

0.70

0.030

(a) Calculate the percent of variability explained by this model.

(b) Calculate Cook’s distance for each observation and provide an interpretation of this statistic.

SOLUTION

The regression equation is

Predictor

Coef

Constant

–0.07837

TEMP

4e-005

(b) Cook's distance values

0.01811

0.00058

0.00191

0.00206

0.02458

0.10379

0.09673

0.01555

Reserve Problems Chapter 12 Section 5 Problem 8

Consider the following stack-loss data from a plant oxidizing ammonia to nitric acid. Twenty-

one daily responses of stack loss (the amount of ammonia escaping) were measured with air flow

1

x

, temperature

2

x

, and acid concentration

3

x

.

y

1

x

2

x

3

x

38

80

27

89

37

80

27

88

41

75

25

90

28

62

24

87

18

62

22

87

18

62

23

87

19

62

24

93

20

62

24

93

19

58

23

87

14

58

18

80

14

58

18

89

13

58

17

88

11

58

18

82

12

58

19

93

12

50

18

89

7

50

18

86

8

50

19

72

8

50

19

79

9

50

20

80

15

56

20

82

15

70

20

91

(a) What proportion of total variability is explained by this model?

(b) Calculate Cook’s distance for the observations in this data set. Are there any influential

points in these data?

SOLUTION

Predictor

Coef

Constant

-43.5

(b) Cook's distance values

Cook’s D

0.00

0.02

0.36

0.13

0.00

0.04

Reserve Problems Chapter 12 Section 5 Problem 9

Heat treating is often used to carburize metal parts such as gears. The thickness of the carburized

layer is considered a crucial feature of the gear and contributes to the overall reliability of the

part. Because of the critical nature of this feature, two different lab tests are performed on each

furnace load. One test is run on a sample pin that accompanies each load. The other test is a

destructive test that cross-sections an actual part. This test involves running a carbon analysis on

the surface of both the gear pitch (top of the gear tooth) and the gear root (between the gear

teeth). Table given below shows the results of the pitch carbon analysis test for 32 parts.

TEMP

SOAKTIME

SOAKPCT

DIFFTIME

DIFFPCT

PITCH

1650

0.58

1.10

0.25

0.90

0.013

1650

0.66

1.10

0.33

0.90

0.018

1650

0.66

1.10

0.33

0.90

0.015

1650

0.66

1.10

0.33

0.95

0.016

1600

0.66

1.15

0.33

1.00

0.015

1600

0.66

1.15

0.33

1.00

0.016

1650

1.00

1.10

0.50

0.80

0.014

1650

1.17

1.10

0.58

0.80

0.021

1650

1.17

1.10

0.58

0.80

0.018

1650

1.17

1.10

0.58

0.80

0.019

1650

1.17

1.10

0.58

0.90

0.021

1650

1.17

1.10

0.58

0.90

0.019

1650

1.17

1.15

0.58

0.90

0.021

1650

1.20

1.15

1.10

0.80

0.025

1650

2.00

1.15

1.00

0.80

0.025

1650

2.00

1.10

1.10

0.80

0.026

1650

2.20

1.10

1.10

0.80

0.024

1650

2.20

1.10

1.10

0.80

0.025

1650

2.20

1.15

1.10

0.80

0.024

1650

2.20

1.10

1.10

0.90

0.025

1650

2.20

1.10

1.10

0.90

0.027

1650

2.20

1.10

1.50

0.90

0.026

1650

3.00

1.15

1.50

0.80

0.029

1650

3.00

1.10

1.50

0.70

0.030

1650

3.00

1.10

1.50

0.75

0.028

1650

3.00

1.15

1.66

0.85

0.032

1650

3.33

1.10

1.50

0.80

0.033

1700

4.00

1.10

1.50

0.70

0.039

1650

4.00

1.10

1.50

0.70

0.041

1650

4.00

1.15

1.50

0.85

0.036

1700

12.50

1.00

1.50

0.70

0.055

1700

18.50

1.00

1.50

0.70

0.068

Fit a model to the response PITCH in the heat-treating data using regressors

1

x

= SOAKTIME

SOAKPCT and

2

x

= DIFFTIME

DIFFPCT.

(a) Calculate the

2

R

for this model.

(b) Calculate Cook’s distance for the observations in this data set. Are there any influential

points in these data?

SOLUTION

Predictor

Coef

Constant

0.010918

(b) Cook's distance values

0.0225580383065276

0.000722191895299444

0.00249387781589604

0.00021380632131336

0.00582268924467583

1.20563909278298e-005

0.0553884923056489

0.0178934129020985

0.00179838440425393

0.000266250330081329

0.00879618259916858

0.000204776558779259

0.00724576249077131

Reserve Supplemental Exercises Chapter 12 Problem 1

A scientist is investigating how the growth rate of a population of animals

( )

y

depends on the

size of population

( )

1

x

and the rate at which the members of this population meet the predators

( )

2

x

. The regressors were measured in specific units. Ten observations were collected, and the

following summary quantities obtained:

10n=

,

1380

i

x=

,

255

i

x=

,

46.1

i

y=

,

2

115762

i

x

=

,

2

2563

i

x

=

,

12 2385

ii

xx=

,

11803

ii

xy=

, and

2185.4

ii

xy=

.

(a) Set up the least squares normal equations for the model

0 1 1 2 2

Y x x

= + + +

.

(b) Estimate the parameters in the model

0 1 1 2 2

Y x x

= + + +ò

.

(c) Determine the fitted value of y when

110x=

and

26.x=

SOLUTION

(a) The least squares normal equations for the model

0 1 1 2 2

Y x x

= + + +ò

:

1 1 1

0 1 1 2 2

ˆ ˆ ˆ

i i i

i i i

n n n

n x x y

= = =

+ + =

Inserting the given summations into the normal equations, we obtain

0 1 2

(b) The least square estimates could be found by solving the system of the normal equations

above or by using the matrix approach:

12

10 380 55

ii

n x x

46.1

y

1.9203

Reserve Supplemental Exercises Chapter 12 Problem 2

A scientist is investigating how the growth rate of a population of animals

( )

y

depends on the

size of population

( )

1

x

and the rate at which the members of this population meet the predators

( )

2

x

. The regressors were measured in specific units. Ten observations were collected, and the

following summary quantities obtained:

10n=

,

1380

i

x=

,

255

i

x=

,

46.1

i

y=

,

2

115762

i

x

=

,

2

2563

i

x

=

,

12 2385

ii

xx=

,

11803

ii

xy=

, and

2185.4

ii

xy=

.

Assume that the total sum of squares for y is

44.6490

T

SS =

.

Use the regression coefficients rounded to at least five decimal places to obtain the answers for

the following questions.

(a) Test for significance of regression using

0.01

=

. Determine the value of test statistic.

Determine whether the regression model is significant or not.

(b) Find the estimate of the error variance.

(c) What is the standard error of the regression coefficient

1

ˆ

?

SOLUTION

(a) The least squares normal equations for the model

0 1 1 2 2

Y x x

= + + +ò

:

12

10 380 55

ii

n x x

46.1

i

y

Regression coefficients:

1.92034

(a)

0 1 2

:0H

==

;

R

SS

Therefore, we reject

0

H

and conclude that the regression model is significant at

0.01

=

.

Reserve Supplemental Exercises Chapter 12 Problem 3

A researcher at a Soap-Bubbles company wants to model the relationship between the quality of

solution for inflating bubbles

( )

y

and the amounts of its four main components: liquid soap

( )

1

x

, sugar

( )

2

x

, glycerol

( )

3

x

and water

( )

4

x

.

The regressors were measured in specific units. The data are shown in the following table.

Liquid soap

Sugar

Glycerol

Water

Solution

4.319

0.226

1.70

3.20

7.128

4.703

0.217

1.34

3.60

7.052

5.172

0.223

1.37

3.55

7.113

4.910

0.219

1.68

3.20

7.098

5.098

0.231

1.56

3.30

7.139

4.841

0.229

1.74

3.10

7.102

4.899

0.227

1.49

3.65

7.048

(a) Fit a multiple linear regression model to this data.

(b) Estimate

2

.

(c) Predict the quality of solution when

14.07x=

,

20.123x=

,

32.0x=

,

43.9x=

.

SOLUTION

Predictor

Coef

SE Coef

T

P

Constant

7.6443

0.9530

8.02

0.015

Analysis of Variance

Reserve Supplemental Exercises Chapter 12 Problem 4

A researcher at a Soap-Bubbles company wants to model the relationship between the quality of

solution for inflating bubbles

( )

y

and the amounts of its four main components: liquid soap

( )

1

x

, sugar

( )

2

x

, glycerol

( )

3

x

and water

( )

4

x

.

The regressors were measured in specific units. The data are shown in the following table.

Liquid soap

Sugar

Glycerol

Water

Solution

4.319

0.226

1.70

3.20

7.128

4.703

0.217

1.34

3.60

7.052

5.172

0.223

1.37

3.55

7.113

4.910

0.219

1.68

3.20

7.098

5.098

0.231

1.56

3.30

7.139

4.841

0.229

1.74

3.10

7.102

4.899

0.227

1.49

3.65

7.048

What is the value of

2

R

for the multiple linear regression model to these data?

SOLUTION

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