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(a)

95% confidence interval on

1

1

(b)

95% confidence interval on

0

(c)

95% confidence interval for the mean rating when the average yards per attempt is 8.0

(d)

Reserve Problems Chapter 11 Sections 5 and 6 Problem 2

The number of pounds of steam used per month by a chemical plant is thought to be related to

the average ambient temperature (in °F) for that month. The past year's usage and temperatures

are in the following table:

Month

Temp.

Usage/1000

Jan.

21

185.79

Feb.

24

214.47

Mar.

32

288.03

Apr.

47

424.84

May

50

454.58

June

59

539.03

July

68

621.55

Aug.

74

675.06

Sept.

62

562.03

Oct.

50

452.93

Nov.

41

369.95

Dec.

30

273.98

Assuming that a simple linear regression model is appropriate, fit the regression model relating

steam usage (y) to the average temperature (x).

(a) Find a 99% confidence interval for

1

.

(b) Find a 99% confidence interval for

0

.

(c) Find a 95% confidence interval on mean steam usage when the average temperature is 51°F.

(d) Find a 95% prediction interval on steam usage when the temperature is 51°F.

SOLUTION

(a)

The regression equation is

(b)

(c)

(d)

Reserve Problems Chapter 11 Sections 5 and 6 Problem 3

The table presents the highway gasoline mileage performance and engine displacement for

DaimlerChrysler vehicles for model year 2005 (U.S. Environmental protection Agency).

Engine Displacement (in3)

MPG (highway)

215

30.8

201

32.5

196

35.4

226

28.1

226

24.4

348

24.1

226

28.5

348

24.2

148

32.8

226

28

122

41.3

215

30.0

215

28.2

148

34.1

500

18.7

348

20.3

165

35.1

148

37.9

148

33.8

500

25.9

148

26.4

Find a 95% confidence interval on each of the following:

a. Slope

b. Intercept

c. Mean highway gasoline mileage when the engine displacement

is x = 159 in3

(d) Construct a 95% prediction interval on highway gasoline mileage when the engine

displacement is x = 159 in3.

SOLUTION

(a)

The regression equation is

(b)

(c)

(d)

33 24

ˆ.1y=

when

159x=

Reserve Problems Chapter 11 Section 7 Problem 1

The following table contains the values of the average annual salary in thousands of dollars (x)

and the food expenses percentage (y) in seven regions of a country.

x

45.1

59.0

57.2

61.8

58.8

47.2

55.2

y

68.8

61.2

59.9

56.7

55.0

54.3

49.3

(a) What proportion of the total variability in the food expenses percentage is accounted for by

the simple linear regression model?

(b) Prepare a normal probability plot of the residuals from the least squares model. Does the

normality assumption seem to be satisfied?

SOLUTION

(a)

The regression equation is

Analysis of Variance

Therefore,

(b)

There is no major departure from the normality assumption in the following graph.

Reserve Problems Chapter 11 Section 7 Problem 2

The following table presents data about the annual sales volume (y in millions of dollars) and the

square footage (x in thousands of square feet) of 14 stores.

x

y

1.8

3.3

1.6

3.7

2.6

6.5

5.1

9.7

0.9

3.3

0.7

5.2

2

3.3

1.2

2.6

3.2

5.3

1.1

2.9

5.4

10.6

4.6

7.5

6.2

11.8

3

4.3

What proportion of the total variability in the sales volume is accounted for by the simple linear

regression model?

SOLUTION

The regression equation is

y = 1.37 + 1.54 x

Analysis of Variance

Therefore,

Reserve Problems Chapter 11 Section 7 Problem 3

It is known that carbon monoxide has a great influence on health. The following data are the

carbon monoxide concentration (x in milligrams per cubic meter) and allergic morbidity (y).

x

y

8.52

14.9

9.73

16.9

8.63

17.2

9.48

16.9

5.98

13.8

9.44

15.3

6.44

15.7

5.71

14.2

4.24

10.7

3.52

11.6

6.23

13.2

(a) Fit the linear regression model by least squares, find the estimates of the slope and intercept.

(b) Find the value of

2

R

.

SOLUTION

(a)

The regression equation is

Analysis of Variance

Source

DF

SS

MS

F

P

(b)

Reserve Problems Chapter 11 Section 7 Problem 4

A company wants to test the effectiveness of their product advertising. It provides different

marketing strategies in 10 regions. The following table contains advertising expenses (x in

millions of dollars) and the number of sales (y in millions of units).

x

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

y

37.5

38.4

38.6

39

41.8

41.2

44.3

45.5

46.6

48.2

(a) Fit the linear regression model by least squares.

(b) Calculate

2

R

for this model.

(c) Estimate

2

for this model.

SOLUTION

(a)

The regression equation is

Predictor

Coef

SE Coef

T

P

Analysis of Variance

Therefore,

(c)

=

Reserve Problems Chapter 11 Section 7 Problem 5

See the table for data on the ratings of quarterbacks for the 2008 National Football League

season (The Sports Network).

Yards per attempt

Rating points

8.39

105.5

7.67

97.4

7.66

96.9

7.98

96.2

7.21

95

7.53

93.8

8.01

92.7

7.66

91.4

7.21

90.2

7.16

89.4

7.93

87.7

7.10

87.5

6.33

87

6.76

86.4

6.86

86.4

7.35

86

7.22

85.4

7.94

84.7

6.41

84.3

6.77

81.7

6.65

81

6.94

80.3

6.45

80.2

7.04

80.1

6.39

79.6

6.58

77.1

6.21

76

7.17

73.7

6.34

72.6

6.18

71.4

5.12

70

5.71

66.5

Assuming that a simple linear regression model is appropriate, fit the regression model relating

the rating (y) to the average number of yards gained per pass attempt (x).

(a) Calculate

2

R

for this model and provide a practical interpretation of this quantity.

The model accounts for _____ percent of the variability in the data.

(b) Select the correct normal probability plot of the residuals from the least squares model.

A

B

C

Does the normality assumption seem to be satisfied?

(c) Look at the plots of residuals versus the fitted values and against x. Interpret these graphs.

SOLUTION

Reserve Problems Chapter 11 Section 7 Problem 6

An article in Wear (1992, Vol. 152, pp. 171-181) presents data on the fretting wear of mild steel

and oil viscosity. Representative data follow with x = oil viscosity and y = wear volume (10-4

cubic millimeters).

y

240

181

193

155

172

110

113

75

94

x

1.6

9.4

15.5

20.0

22.0

35.5

43.0

40.5

33.0

(a) Calculate

2

R

for the simple linear regression model and provide an interpretation of this

quantity. The model accounts for _____ percent of the variability in the data.

(b) Look at the plots of the residuals from this model versus

ˆ

y

and versus x. Interpret these plots.

(c) Select the corret normal probability plot of the residuals.

A

B

C

Does the normality assumption appear to be satisfied?

SOLUTION

(a)

The assumption of constant variance appears reasonable.

(c)

Reserve Problems Chapter 11 Section 7 Problem 7

A rocket motor is manufactured by bonding together two types of propellants, an igniter and a

sustainer. The shear strength of the bond y is thought to be a linear function of the age of the

propellant x when the motor is cast. The following table provides 20 observations.

Observation

number

Strength y (psi)

Age x (weeks)

1

2158.70

15.50

2

1678.15

23.75

3

2316.00

8.00

4

2061.30

17.00

5

2207.50

5.00

6

1708.30

19.00

7

1784.70

24.00

8

2575.00

2.50

9

2357.90

7.50

10

2277.70

11.00

11

2165.20

13.00

12

2399.55

3.75

13

1779.80

25.00

14

2336.75

9.75

15

1765.30

22.00

16

2053.50

18.00

17

2414.40

6.00

18

2200.50

12.50

19

2654.20

2.00

20

1753.70

21.50

(a) Calculate

2

R

for

the simple linear regression model and provide

an interpretation of this quantity.

The model accounts for ____

percent

of the variability in the data.

(b) Select the correct graph of the residuals on a normal probability scale.

A

B

C

Do any points seem unusual on this plot?

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