doe hw 11-26_copy (1)

8/10/2019 DOE HW 11-26_Copy (1)

1/14

Question 10.15

A University career center conducted a study to determine whether there is an association

between starting salaries Y (in thousands of dollors) and grade point average X1,

age apon completion X2, and gender for students in the school of engineering.

The career center obtained the following sample data

Y X1 X2 X3Starting Salary GPA Age Gender

50.5 2.95 22 F

52 3.2 23 M

53.1 3.4 22 M

54.4 3.6 23 M

53.2 3.5 24 M

47 2.85 24 F

50 3.1 25 F

50.8 3.2 26 F

47.7 3.05 23 M

46.4 2.7 24 F

47.5 2.75 28 F

49.2 3.1 22 M

51 3.15 22 M

49.2 2.95 23 F

48.8 0.75 26 M

(a) Fit an appropriate regression model to these data, evaluate it

and reive it as suggested by your evaluation

(b) Think of another potential predictor variable that could further

explain the variation in the sample starting salaries

Solution

(a)

Y X1 X2 X3

50.5 2.95 22 0

52 3.2 23 1

53.1 3.4 22 1

54.4 3.6 23 1

53.2 3.5 24 1

47 2.85 24 0

50 3.1 25 050.8 3.2 26 0

47.7 3.05 23 1

46.4 2.7 24 0

47.5 2.75 28 0

49.2 3.1 22 1

51 3.15 22 1

49.2 2.95 23 0

48.8 0.75 26 1

8/10/2019 DOE HW 11-26_Copy (1)

2/14

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.703494178

R Square 0.494904058Adjusted R S 0.35715062

Standard Erro 1.931858531

Observations 15

ANOVA

df SS MS F

Significa

nce F

Regression 3 40.2245 13.4082 3.59268 0.04982

Residual 11 41.0529 3.73208

Total 14 81.2773

Coefficients Standard Error t Stat P-value

Lower

95%

Upper

95%

Lower

95.0%

Upper

95.0%

Intercept 40.3640753 10.34532195 3.90167 0.00247 17.59418 63.134 17.5942 63.134

X1 1.8664914 0.883585363 2.11241 0.05833 -0.07827 3.81125 -0.07827 3.81125

X2 0.11969999 0.360596758 0.33195 0.74617 -0.67397 0.91337 -0.67397 0.91337

X3 2.50171596 1.120612565 2.23245 0.04732 0.035264 4.96817 0.03526 4.96817

Y X1 X3

50.5 2.95 0

52 3.2 1

53.1 3.4 1

54.4 3.6 1

53.2 3.5 1

47 2.85 0

50 3.1 0

50.8 3.2 0

47.7 3.05 1

46.4 2.7 047.5 2.75 0

49.2 3.1 1

51 3.15 1

49.2 2.95 0

48.8 0.75 1

8/10/2019 DOE HW 11-26_Copy (1)

3/14

SUMMARY OUTPUT


Multiple R 0.69989

R Square 0.48984

Adjusted R Squa 0.40482Standard Error 1.85885

Observations 15

ANOVA

df SS MS F

Significa

nce F

Regression 2 39.8132 19.9066 5.761116 0.01763

Residual 12 41.4641 3.45534

Total 14 81.2773

CoefficientsStandardError t Stat P-value

Lower95%

Upper95%

Lower95.0%

Uppe95.0%

Intercept 43.70409143 2.31443 18.8833 2.73E-10 38.6614 48.7468 38.6614 48.74

X1 1.730310245 0.753 2.29789 0.040352 0.08966 3.37096 0.08966 3.370

X3 2.334050035 0.96252 2.42493 0.032028 0.23689 4.43121 0.23689 4.431

The Model would be:

Y=43.704+1.73031X1+2.33405X2

8/10/2019 DOE HW 11-26_Copy (1)

4/14

Question 10.17

An insurance executive wished to estimate the relationship between the number of days

of work lost by auto accident victims Y and age X1 and gender X2 of victim. A representative

sample of 25 loss reports was selected resulting in the least squares equation

Y-Bar=21.4-0.0072X1-2.5X2, For this equation SST = 4.750, SSE=3.180, SD(b1)=0.11, SD(b

(a) Do you detect an association between the response variable Y and the two predictor

variables as a group? Support your answer(b) Is the incremental contribution of age discernible, given the persons gender? Explain

(c ) Is the incremental contribution of gender discernible, given the persons age? Explain

(d) what do your conclusions in parts (a)-(c ) suggest about basing premiums for income

replacement on the age and gender of the unsured when work time is lost due to an

automobile accident?

Solution:

Y-Bar=21.4-0.0072X1-2.5X2

(a) the data sample for analysis can be considered as

(X2: 0=Female, 1= Male)

X3 = X1*X2

Y X1 X2 X3

21.2416 22 0 0

18.7344 23 1 23

18.7416 22 1 22

18.7344 23 1 23

18.7272 24 1 2421.2272 24 0 0

21.22 25 0 0

21.2128 26 0 0

18.7344 23 1 23

21.2272 24 0 0

21.1984 28 0 0

18.7416 22 1 22

18.7416 22 1 22

21.2344 23 0 0

18.7128 26 1 26

Y X3

21.2416 0

18.7344 23

18.7416 22

18.7344 23

8/10/2019 DOE HW 11-26_Copy (1)

5/14

18.7272 24

21.2272 0

21.22 0

21.2128 0

18.7344 23

21.2272 0

21.1984 018.7416 22

18.7416 22

21.2344 0

18.7128 26

SUMMARY OUTPUT


Multiple R 0.997180093R Square 0.994368138

Adjusted R Sq 0.993934918

Standard Error 0.100126163

Observations 15

ANOVA

df SS MS

Regression 1 23.0109068 23.0109068

Residual 13 0.130328232 0.01002525

Total 14 23.14123503

Coefficients Standard Error t Stat P-value

Intercept 21.21514683 0.037779413 561.5531057 6.8402E-30

X3 -0.107014068 0.002233683 -47.90924109 5.211E-16

F

2295.295382

The model can be defined as

Lower 95%21.13352937

-0.111839647

(b) Considering incremental contribution of age against given gender

Y X1 X2

21.2416 22 0

18.7344 22 1

Y=21.21515-0.10701X3

8/10/2019 DOE HW 11-26_Copy (1)

6/14

8/10/2019 DOE HW 11-26_Copy (1)

7/14

18.7416 22 0 0.354433637

18.7344 23 0 4.82175E-26

18.7272 24 0

21.2272 24 0

21.22 25 0

21.2128 26 1

18.7344 23 121.2272 24 1

21.1984 28 1

18.7416 22 1

18.7416 22 1

21.2344 23 1

18.7128 26 1

SUMMARY OUTPUT


Multiple R 0.413975326

R Square 0.171375571

Adjusted R Sq 0.033271499


Observations 15

ANOVA

df SS MS

Regression 2 3.965842359 1.98292118Residual 12 19.17539267 1.59794939

Total 14 23.14123503

Coefficients Standard Error t Stat

Intercept 12.6565445 4.625799429 2.73607723

X1 0.306936126 0.197591481 1.55338744

X2 -0.12434555 0.681414417 -0.1824815 F

1.240916135

P-value

0.018062093

0.146295239

0.858251378

8/10/2019 DOE HW 11-26_Copy (1)

8/14

)=0.99

8/10/2019 DOE HW 11-26_Copy (1)

9/14

Significance F

5.21105E-16

Upper 95% Lower 95.0% Upper 95.0%21.29676429 21.13352937 21.29676429

-0.102188488 -0.111839647 -0.102188488

8/10/2019 DOE HW 11-26_Copy (1)

10/14

8/10/2019 DOE HW 11-26_Copy (1)

11/14

-0.005756376 0.002226964 -0.005756376 0.00223

-2.502595094 -2.47506373 -2.502595094 -2.47506

Significance F

0.323702756

Lower 95% Upper 95% Lower 95.0%

Upper

95.0%

2.57779336 22.73529565 2.57779336 22.7353

-0.123578729 0.73745098 -0.123578729 0.73745

-1.609020024 1.360328925 -1.609020024 1.36033

8/10/2019 DOE HW 11-26_Copy (1)

12/14

Question 10.23

A Manufacturing firm wishes to predict the manufacturing unit cost Y (in dollors) of one

of its products as a function of fluctuating production rate X1, and material and labor costs X2,

(X1 is measured as a percentage of rated capacity and X2 is a standard index that combines

the costs of material and labor.) Representative data were collected over a 20 month span

during which the production rate and labor costs fluctuated considerably

Y X1 X2 Y X1 X2

13.59 87 80 15.93 102 116

15.71 78 95 16.45 82 117

15.97 81 106 19.02 74 127

20.21 65 115 18.16 85 133

24.64 51 128 18.57 86 135

21.25 62 128 17.01 90 136

18.94 70 115 18.03 93 140

14.85 91 92 19.22 81 142

15.18 94 93 21.12 72 14816.3 100 111 23.32 60 150

Fit an appropriate regression model to these data, evaluate the resulting least squares

equation, and revise it as necessary

Solution

Month # Y X1 X2

1 13.59 87 80

2 15.71 78 953 15.97 81 106

4 20.21 65 115

5 24.64 51 128

6 21.25 62 128

7 18.94 70 115

8 14.85 91 92

9 15.18 94 93

10 16.3 100 111

11 15.93 102 116

12 16.45 82 11713 19.02 74 127

14 18.16 85 133

15 18.57 86 135

16 17.01 90 136

17 18.03 93 140

18 19.22 81 142

19 21.12 72 148

8/10/2019 DOE HW 11-26_Copy (1)

13/14

20 23.32 60 150

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.95601

R Square 0.91396

Adjusted R Square 0.90384


Observations 20

ANOVA

df SS MS F

Significa

nce F

Regression 2 144.387 72.1937 90.28915 8.8E-10Residual 17 13.5929 0.79958

Total 19 157.98

Coefficients

Standard

Error t Stat P-value

Lower

95%

Upper

95%

Lower

95.0%

Upper

95.0%

Intercept 20.28126941 2.12525 9.543 3.1E-08 15.79738 24.7652 15.7974 24.7652

X1 -0.137695838 0.01585 -8.68549 1.2E-07 -0.17114 -0.10425 -0.17114 -0.10425

X2 0.074245424 0.01096 6.77134 3.3E-06 0.051112 0.09738 0.05111 0.09738

Model Would be

Y=20.2812-0.1376X1+0.07424X2

8/10/2019 DOE HW 11-26_Copy (1)

14/14

10.21

How well can a taxpayer's taxes Y, as a percentage of his or her gross income X (in thousands

of dollors)? The following represents a random sample of 14 federal income tax returns in a

given year"

Income X % tax Y:

45.6 10.462.2 11.8

77.6 14.7

118.8 16.7

30.4 5.8

50.1 10.2

60 13.9

49.3 10.9

36.1 7

38 9.1

108.2 16.1

54 12.6

42.1 9.8

90 16.6

Fit a simple linear regression model to these data, evaluate the resulting least squares

equation (including a residual analysis) and revise it as necessary.

Linear Regression Equations:

y = 0.1157x + 4.7037

R = 0.8363

% tax Y:

0

5

10

15

20

% tax Y:

doe hw 11-26_copy (1)

Documents