# Statistics II Week 6 Homework | Complete Solution

8.4 Elasticity of moissanite. Moissanite is popular abrasive material because of its extreme hardness. Another important property of moissanite is elasticity. The elastic properties of the material were investigated in the Journal of Applied Physics (September 1993). A diamond anvil cell was used to compress a mixture of moissanite, sodium chloride, and gold in a ratio of 33.99:1 by volume. The compressed volume, y, of the mixture (relative to the zero-pressure volume) was measured at each of 11 different pressures (GPa). The results are displayed in the table (p.397). A MINITAB printout for the straight-line regression model E(y)= β_0+ β_1 x and a MINITAB residual plot are displayed at left.

MOISSANITE

COMPRESSED VOLUME y, % PRESSURE x, GPa

100 0

96 9.4

93.8 15.8

90.2 30.4

87.7 41.6

86.2 46.9

85.2 51.6

83.3 60.1

82.9 62.6

82.9 62.6

81.7 68.4

Calculate the regression residuals.

Plot the residuals against x. Do you detect a trend?

Propose an alternative model based on the plot part b.

Fit and analyze the model you proposed in part c.

8.12 Fair market value of Hawaiian properties. Prior to 1980, private homeowners in Hawaii had to lease the land their homes were built on because the law (dating back to the islands’ feudal period) required that land be owned only by the big estates. After 1980, however, a new law instituted condemnation proceedings so that citizens could buy their own land. To comply with the 1980 law, one large Hawaiian estate wanted to use regression analysis to estimate the fair market value of its land. Its first proposal was the quadratic model

E(y)= β_0+ β_1 x+ β_2 x^2

HAWAII

PROPERTY LEASED FEE VALUE

y, thousands of dollars SIZE

x, thousands

1 70.7 13.5

2 52.7 9.6

3 87.6 17.6

4 43.2 7.9

5 103.8 11.5

6 45.1 8.2

7 86.8 15.2

8 73.3 12.0

9 144.3 13.8

10 61.3 10.0

11 148.0 14.5

12 85.0 10.2

13 171.2 18.7

14 97.5 13.2

15 158.1 16.3

16 74.2 12.3

17 47.0 7.7

18 54.7 9.9

19 68.0 11.2

20 75.2 12.4

Where

y=Leased fee value (i.e.,sale price of property)

x=Size of property in square feet

Data collected for 20 property sales in a particular neighborhood, given in the table above, were used to fit the model. The least squares prediction equation is

ŷ=-44.0947+11.5339x-.06378x^2

Calculate the predicted values and corresponding residuals for the model.

Plot the residuals versus ŷ . Do you detect any trends? If so what does the pattern suggest about the model?

Conduct a test for heteroscedasticity. [Hint: Divide the data into two subsamples, x≤12 and x 12, and fit the model to both subsamples.]

Based on your results from parts b and c, how should the estate proceed?

8.20 Cooling method for gas turbines. Refer to the Journal of Egineering for Gas Turbines and Power (January 2005) study of a high-pressure inlet fogging method for a gas turbine engine, Exercise 8.11 (p.407). Use a residual graph to check the assumption of normal errors for the interaction model for heat rate (y). Is the normality assumption reasonably satisfied? If not suggest how to modify the model.

8.28 Modeling an employee’s work-hours missed. A large manufacturing firm wants to determine whether a relationship exists between y, the number of work-hours an employee misses per year, and x, the employee’s annual wages. A sample of 15 employees produced the data in the accompanying table.

MISSWORK

EMPLOYEE WORK-HOURS MISSED y ANNUAL WAGES x, thousands of dollars

1 49 12.8

2 36 14.5

3 127 8.3

4 91 10.2

5 72 10.0

6 34 11.5

7 155 8.8

8 11 17.2

9 191 7.8

10 6 15.8

11 63 10.8

12 79 9.7

13 543 12.1

14 57 21.2

15 82 10.9

Fit the first-order model, (y)= β_0+ β_1 x , to the data.

Plot the regression residuals. What do you notice?

After searching through its employees’ files, the firm found that employee #13 had been fired but that his name had not been removed from the active employee payroll. This explains the large accumulation of work-hours missed (543) by that employee. In view of his fact, what is your recommendation concerning this outlier?

8.46 Analysis of television market share. The data in the table are the monthly market shares for a product over most of the past year. The least squares line relating market share to television advertising expenditure is found to be

ŷ=-1.56+.687x

TVSHARE

MONTH MARKET SHARE

y,% TELEVISION ADVERTISING EXPENDITURE

x, thousands of dollars

January 15 23

February 17 27

March 17 25

May 13 21

June 12 20

July 14 24

September 16 26

October 14 23

December 15 25

Calculate and plot the regression residuals in the manner outlined in this section.

The response variable y, market share, is recorded as a percentage. What does this lead you to believe about the least squares assumption of homoscedasticity? Does the residual plot substantiate this belief?

What variance-stabilizing transformation is suggested by the trend in the residual plot? Refit the first-order model using the transformed responses. Calculate and plot these new regression residuals. Is there evidence that the transformation has been successful in stabilizing the variance of the error term, ε?

MOISSANITE

COMPRESSED VOLUME y, % PRESSURE x, GPa

100 0

96 9.4

93.8 15.8

90.2 30.4

87.7 41.6

86.2 46.9

85.2 51.6

83.3 60.1

82.9 62.6

82.9 62.6

81.7 68.4

Calculate the regression residuals.

Plot the residuals against x. Do you detect a trend?

Propose an alternative model based on the plot part b.

Fit and analyze the model you proposed in part c.

8.12 Fair market value of Hawaiian properties. Prior to 1980, private homeowners in Hawaii had to lease the land their homes were built on because the law (dating back to the islands’ feudal period) required that land be owned only by the big estates. After 1980, however, a new law instituted condemnation proceedings so that citizens could buy their own land. To comply with the 1980 law, one large Hawaiian estate wanted to use regression analysis to estimate the fair market value of its land. Its first proposal was the quadratic model

E(y)= β_0+ β_1 x+ β_2 x^2

HAWAII

PROPERTY LEASED FEE VALUE

y, thousands of dollars SIZE

x, thousands

1 70.7 13.5

2 52.7 9.6

3 87.6 17.6

4 43.2 7.9

5 103.8 11.5

6 45.1 8.2

7 86.8 15.2

8 73.3 12.0

9 144.3 13.8

10 61.3 10.0

11 148.0 14.5

12 85.0 10.2

13 171.2 18.7

14 97.5 13.2

15 158.1 16.3

16 74.2 12.3

17 47.0 7.7

18 54.7 9.9

19 68.0 11.2

20 75.2 12.4

Where

y=Leased fee value (i.e.,sale price of property)

x=Size of property in square feet

Data collected for 20 property sales in a particular neighborhood, given in the table above, were used to fit the model. The least squares prediction equation is

ŷ=-44.0947+11.5339x-.06378x^2

Calculate the predicted values and corresponding residuals for the model.

Plot the residuals versus ŷ . Do you detect any trends? If so what does the pattern suggest about the model?

Conduct a test for heteroscedasticity. [Hint: Divide the data into two subsamples, x≤12 and x 12, and fit the model to both subsamples.]

Based on your results from parts b and c, how should the estate proceed?

8.20 Cooling method for gas turbines. Refer to the Journal of Egineering for Gas Turbines and Power (January 2005) study of a high-pressure inlet fogging method for a gas turbine engine, Exercise 8.11 (p.407). Use a residual graph to check the assumption of normal errors for the interaction model for heat rate (y). Is the normality assumption reasonably satisfied? If not suggest how to modify the model.

8.28 Modeling an employee’s work-hours missed. A large manufacturing firm wants to determine whether a relationship exists between y, the number of work-hours an employee misses per year, and x, the employee’s annual wages. A sample of 15 employees produced the data in the accompanying table.

MISSWORK

EMPLOYEE WORK-HOURS MISSED y ANNUAL WAGES x, thousands of dollars

1 49 12.8

2 36 14.5

3 127 8.3

4 91 10.2

5 72 10.0

6 34 11.5

7 155 8.8

8 11 17.2

9 191 7.8

10 6 15.8

11 63 10.8

12 79 9.7

13 543 12.1

14 57 21.2

15 82 10.9

Fit the first-order model, (y)= β_0+ β_1 x , to the data.

Plot the regression residuals. What do you notice?

After searching through its employees’ files, the firm found that employee #13 had been fired but that his name had not been removed from the active employee payroll. This explains the large accumulation of work-hours missed (543) by that employee. In view of his fact, what is your recommendation concerning this outlier?

8.46 Analysis of television market share. The data in the table are the monthly market shares for a product over most of the past year. The least squares line relating market share to television advertising expenditure is found to be

ŷ=-1.56+.687x

TVSHARE

MONTH MARKET SHARE

y,% TELEVISION ADVERTISING EXPENDITURE

x, thousands of dollars

January 15 23

February 17 27

March 17 25

May 13 21

June 12 20

July 14 24

September 16 26

October 14 23

December 15 25

Calculate and plot the regression residuals in the manner outlined in this section.

The response variable y, market share, is recorded as a percentage. What does this lead you to believe about the least squares assumption of homoscedasticity? Does the residual plot substantiate this belief?

What variance-stabilizing transformation is suggested by the trend in the residual plot? Refit the first-order model using the transformed responses. Calculate and plot these new regression residuals. Is there evidence that the transformation has been successful in stabilizing the variance of the error term, ε?

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