Statistics 8th Ed by Newbold Solutions 12 PDF

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Chapter 12:12-2 Statistics for Business & Economics, 8th edition12 The estimated regression slope coefficients are interpreted as follows: b 1 = .661: All else being equal, an increase in the plane’s top speed by one mph will increase the expected number of hours in the design effort by an e...

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Chapter 12: Multiple Regression

12.1 Given the following estimated linear model: yˆ 10  3 x1  2 x2  4 x3 a. yˆ 10  3(20)  2(11)  4(10) = 132 b. yˆ 10  3(15)  2(14)  4(20) = 163 c. ˆy 10 3(35)  2(19)  4(25) = 253 d. ˆy 10 3(10)  2(17)  4(30) = 194 12.2 Given the following estimated linear model: yˆ 10  5 x1  4 x2  2 x3 a. yˆ 10  5(20)  4(11)  2(10) = 174 b. yˆ 10  5(15)  4(14)  2(20) = 181 c. yˆ 10  5(35)  4(19)  2(25) = 311 d. yˆ 10  5(10)  4(17)  2(30) = 188 12.3 Given the following estimated linear model: yˆ 10  2 x1 12 x2  8 x3 a. yˆ 10  2(20) 12(11)  8(10) = 262 b. yˆ 10  2(15) 12(24)  8(20) = 488 c. ˆy 10  2(20) 12(19)  8(25) = 478 d. yˆ 10  2(10) 12(9)  8(30) = 378 12.4 Given the following estimated linear model: yˆ 10  2 x1 12 x2  8 x3 a. ˆy increases by 8 b. ˆy increases by 8 c. ˆy increases by 24 12.5 Given the following estimated linear model: yˆ 10  2 x1  14 x2  6 x3 a. ˆy decreases by 8 b. ˆy decreases by 6 c. ˆy increases by 28

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12-1

12-2 12.6

Statistics for Business & Economics, 8th edition

The estimated regression slope coefficients are interpreted as follows: b1 = .661: All else being equal, an increase in the plane’s top speed by one mph will increase the expected number of hours in the design effort by an estimated . 661 million or 661 thousand worker-hours. b2 = .065: All else being equal, an increase in the plane’s weight by one ton will increase the expected number of hours in the design effort by an estimated .065 million or 65 thousand worker-hours. b3 = -.018: All else being equal, an increase in the percentage of parts in common with other models will result in a decrease in the expected number of hours in the design effort by an estimated .018 million or 18 thousand worker-hours.

12.7

The estimated regression slope coefficients are interpreted as follows: b1 = .057: All else being equal, an increase of one unit in the change over the quarter in bond purchases by financial institutions results in an estimated .057 increase in the change over the quarter in the bond interest rates. b2 = -.065: All else being equal, an increase of one unit in the change over the quarter in bond sales by financial institutions results in an estimated .065 decrease in the change over the quarter in the bond interest rates.

12.8

a. b1 = .052: All else being equal, an increase of one hundred dollars in weekly income results in an estimated .052 quart per week increase in milk consumption. b 2 = 1.14: All else being equal, an increase in family size by one person will result in an estimated increase in milk consumption by 1.14 quarts per week. b. The intercept term b0 of -.025 is the estimated milk consumption of quarts of milk per week given that the family’s weekly income is 0 dollars and there are 0 members in the family. This is likely extrapolating beyond the observed data series and is not a useful interpretation.

12.9

a. b1 = .653: All else being equal, a one unit increase in the average number of meals eaten per week will result in an estimated .653 pound gained during freshman year. b2 = -1.345: All else being equal, a one unit increase in the average number of hours of exercise per week will result in an estimated 1.345 pound weight loss. b3 = .613: All else being equal, a one unit increase in the average number of beers consumed per week will result in an estimated .613 pound weight gain. b. The intercept term b0 of 7.35 is the estimated amount of weight gain during the freshman year given that the meals eaten is 0, hours exercise is 0 and there are no beers consumed per week. This is likely extrapolating beyond the observed data series and is not a useful interpretation.

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Chapter 12: Multiple Regression

12-3

12.10 Compute the slope coefficients for the model: yˆi b0  b1 x1 i  b2 x2 i sy ( rx1y  rx1x 2 rx 2y ) s y ( rx2 y  rx1x2 rx1y ) Given that b1  , b2  2 2 sx1 (1  r x1x 2 ) sx 2 (1 r x 1x 2 ) 400(.60  (.50)(.70)) = .667, 200(1  .50 2 ) 400(.70  (.50)(.60)) b2  = 2.133 100(1  .50 2 ) 400( .60  ( .50)(.70)) = -.667, b. b1  200(1  (  .50) 2 ) 400(.70  ( .50)(  .60)) b2  = 2.133 100(1  (  .50) 2) 400(.40  (.80)(.45)) c. b1  =.222, 200(1  (.80) 2 ) 400(.45  (.80)(.40)) b2  = 1.444 100(1  (.80)2 ) 400(.60  ( .60)(  .50)) = .9375, d. b1  200(1  (  .60) 2 ) 400(  .50  (  .60)(.60)) b2  = -.875 100(1  (  .60) 2) a. b1 

12.11 a. For Y = a0 + a1X1,

.

When the correlation between X1 and X2 is 0, a1 is still defined as,

. This

is because there is no X2 in the equation, Y = a0 + a1X1. When the correlation between X1 and X2 is 0, then the slope coefficient of the X1 term in Y = b0 + b1X1 + b2X2 simplifies to the slope coefficient of the bivariate regression: sy ( rx1y  rx1x 2 rx 2y ) Start with equation 12.4: b1  . 2 sx1 (1  r x1x 2 ) Note that if the correlation between X 1 and X2 is zero, then the second terms in both the numerator and denominator are zero and the formula algebraically sy rx1 y reduces to b1  which is the equivalent of the bivariate slope coefficient. sx1 b. For Y = a0 + a1X1, when the correlation between X1 and X2 is 1, a1 is still defined as,

.

For Y = b0 + b1X1 + b2X2,when the correlation between X1 and X2 is 1, then the denominator goes to 0 and the slope coefficient is undefined.

12-4

Statistics for Business & Economics, 8th edition

12.12 a. Electricity sales as a function of number of customers and price Regression Analysis: SalesMwh versus Pricelec, Numcust The regression equation is SalesMwh = - 584616 + 15421 Pricelec + 2.31 Numcust Predictor Constant Pricelec Numcust

Coef -584616 15421 2.3068

S = 64625.1

SE Coef 267660 21103 0.2007

R-Sq = 81.6%

T -2.18 0.73 11.49

P 0.033 0.468 0.000

R-Sq(adj) = 81.0%

Analysis of Variance Source Regression Residual Error Total

DF 2 65 67

SS 1.20344E+12 2.71466E+11 1.47490E+12

MS 6.01719E+11 4176402085

F 144.08

P 0.000

All else being equal, for every one unit increase in the price of electricity, we estimate that sales will increase by 15,421 mwh. Note that this estimated coefficient is not significantly different from zero ( p-value = .468). All else being equal, for every additional residential customer who uses electricity in the heating of their home, we estimate that sales will increase by 2.31 mwh. b. Electricity sales as a function of number of customers Regression Analysis: salesmw2 versus numcust2 The regression equation is salesmw2 = - 410202 + 2.20 numcust2 Predictor Coef SE Coef Constant -410202 114132 numcust2 2.2027 0.1445 S = 66282 R-Sq Analysis of Variance Source DF Regression 1 Residual Error 62 Total 63

= 78.9%

T -3.59 15.25

P 0.001 0.000

R-Sq(adj) = 78.6%

SS MS 1.02136E+12 1.02136E+12 2.72381E+11 4393240914 1.29374E+12

F 232.48

P 0.000

Regression Analysis: SalesMwh versus Numcust The regression equation is SalesMwh = - 404100 + 2.19 Numcust Predictor Constant Numcust

Coef -404100 2.1947

S = 64396.5

SE Coef 102674 0.1290

R-Sq = 81.4%

T -3.94 17.02

P 0.000 0.000

R-Sq(adj) = 81.2%

Analysis of Variance Source Regression Residual Error Total

DF 1 66 67

SS 1.20121E+12 2.73696E+11 1.47490E+12

MS 1.20121E+12 4146912960

F 289.66

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P 0.000

Chapter 12: Multiple Regression

12-5

An additional residential customer will add 2.19 mwh to electricity sales. The two models have roughly equivalent explanatory power; therefore, adding price as a variable does not add a significant amount of explanatory power to the model. There appears to be a problem of high correlation between the independent variables of price and customers. c. Electricity sales as a function of price and degree days Regression Analysis: SalesMwh versus Pricelec, Degreday The regression equation is SalesMwh = 2370438 - 173882 Pricelec + 56.7 Degreday Predictor Constant Pricelec Degreday S = 111735

Coef 2370438 -173882 56.69

SE Coef 142346 23870 58.65

R-Sq = 45.0%

T 16.65 -7.28 0.97

P 0.000 0.000 0.337

R-Sq(adj) = 43.3%

Analysis of Variance Source Regression Residual Error Total

DF 2 65 67

SS 6.63398E+11 8.11506E+11 1.47490E+12

MS 3.31699E+11 12484705377

F 26.57

P 0.000

All else being equal, an increase in the price of electricity will reduce electricity sales by 173,882 mwh. All else being equal, an increase in the degree days (departure from normal weather) by one unit will increase electricity sales by 56.7 mwh. Note that the coefficient on the price variable is now negative, as expected, and it is significantly different from zero ( p-value = .000)

12-6

Statistics for Business & Economics, 8th edition

d. Electricity sales as a function of disposable income and degree days Regression Analysis: SalesMwh versus YD87, Degreday The regression equation is SalesMwh = 317513 + 318 YD87 + 57.1 Degreday Predictor Constant YD87 Degreday

Coef 317513 317.91 57.06

S = 64613.5

SE Coef 60649 18.73 33.70

R-Sq = 81.6%

T 5.24 16.97 1.69

P 0.000 0.000 0.095

R-Sq(adj) = 81.0%

Analysis of Variance Source Regression Residual Error Total

DF 2 65 67

SS 1.20353E+12 2.71369E+11 1.47490E+12

MS 6.01767E+11 4174903356

F 144.14

P 0.000

All else being equal, an increase in personal disposable income by one unit will increase electricity sales by 318 mwh. All else being equal, an increase in degree days by one unit will increase electricity sales by 57.1 mwh. 12.13 a. mpg as a function of horsepower and weight Regression Analysis: milpgal versus horspwr, weight The regression equation is milpgal = 55.8 - 0.105 horspwr - 0.00661 weight 150 cases used 5 cases contain missing values Predictor Coef SE Coef T P Constant 55.769 1.448 38.51 0.000 horspwr -0.10489 0.02233 -4.70 0.000 weight -0.0066143 0.0009015 -7.34 0.000 S = 3.901 R-Sq = 72.3% R-Sq(adj) = 72.0% Analysis of Variance Source DF SS MS F Regression 2 5850.0 2925.0 192.23 Residual Error 147 2236.8 15.2 Total 149 8086.8

P 0.000

All else being equal, a one unit increase in the horsepower of the engine will reduce fuel mileage by .10489 mpg. All else being equal, an increase in the weight of the car by 100 pounds will reduce fuel mileage by .66143 mpg. b. Add number of cylinders Regression Analysis: milpgal versus horspwr, weight, cylinder The regression equation is milpgal = 55.9 - 0.117 horspwr - 0.00758 weight 150 cases used 5 cases contain missing values Predictor Coef SE Coef T Constant 55.925 1.443 38.77 horspwr -0.11744 0.02344 -5.01 weight -0.007576 0.001066 -7.10 cylinder 0.7260 0.4362 1.66 S = 3.878 R-Sq = 72.9% Analysis of Variance Source DF

+ 0.726 cylinder P 0.000 0.000 0.000 0.098

R-Sq(adj) = 72.3% SS

MS

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F

P

Chapter 12: Multiple Regression

Regression Residual Error Total

3 146 149

5891.6 2195.1 8086.8

1963.9 15.0

130.62

12-7

0.000

All else being equal, one additional cylinder in the engine of the auto will increase fuel mileage by .726 mpg. Note that this is not significant at the .05 level (p-value = .098). Horsepower and weight still have the expected negative signs c. mpg as a function of weight, number of cylinders Regression Analysis: milpgal versus weight, cylinder The regression equation is milpgal = 55.9 - 0.0104 weight + 0.121 cylinder 154 cases used 1 cases contain missing values Predictor Coef SE Coef T P Constant 55.914 1.525 36.65 0.000 weight -0.0103680 0.0009779 -10.60 0.000 cylinder 0.1207 0.4311 0.28 0.780 S = 4.151 R-Sq = 68.8% R-Sq(adj) = 68.3% Analysis of Variance Source DF SS MS F Regression 2 5725.0 2862.5 166.13 Residual Error 151 2601.8 17.2 Total 153 8326.8

P 0.000

All else being equal, an increase in the weight of the car by 100 pounds will reduce fuel mileage by 1.0368 mpg. All else being equal, an increase in the number of cylinders in the engine will increase mpg by .1207 mpg. The explanatory power of the models has stayed relatively the same with a slight drop in explanatory power for the latest regression model. Note that the coefficient on weight has stayed negative and significant (p-values of .000) for all of the regression models; although the value of the coefficient has changed. The number of cylinders is not significantly different from zero in either of the models where it was used as an independent variable. There is likely some correlation between cylinders and the weight of the car as well as between cylinders and the horsepower of the car. d. mpg as a function of horsepower, weight, price Regression Analysis: milpgal versus horspwr, weight, price The regression equation is milpgal = 54.4 - 0.0938 horspwr - 0.00735 weight +0.000137 price 150 cases used 5 cases contain missing values Predictor Coef SE Coef T P Constant 54.369 1.454 37.40 0.000 horspwr -0.09381 0.02177 -4.31 0.000 weight -0.0073518 0.0008950 -8.21 0.000 price 0.00013721 0.00003950 3.47 0.001 S = 3.762 R-Sq = 74.5% R-Sq(adj) = 73.9% Analysis of Variance Source DF SS MS F Regression 3 6020.7 2006.9 141.82 Residual Error 146 2066.0 14.2 Total 149 8086.8

P 0.000

All else being equal, an increase by one unit in the horsepower of the auto will reduce fuel mileage by .09381 mpg. All else being equal, an increase by 100 pounds in the

12-8

Statistics for Business & Economics, 8th edition

weight of the auto will reduce fuel mileage by .73518 mpg and an increase in the price of the auto by one dollar will increase fuel mileage by .00013721 mpg. e. Horse power and weight remain significant negative independent variables throughout whereas the number of cylinders has been insignificant. The size of the coefficients change as the combinations of independent variables changes. This is likely due to strong correlation that may exist between the independent variables. 12.14

a. Horsepower as a function of weight, cubic inches of displacement Regression Analysis: horspwr versus weight, displace The regression equation is horspwr = 23.5 + 0.0154 weight + 0.157 displace 151 cases used 4 cases contain missing values Predictor Coef SE Coef T P Constant 23.496 7.341 3.20 0.002 weight 0.015432 0.004538 3.40 0.001 displace 0.15667 0.03746 4.18 0.000 S = 13.64 R-Sq = 69.2% R-Sq(adj) = 68.8% Analysis of Variance Source DF SS MS F Regression 2 61929 30964 166.33 Residual Error 148 27551 186 Total 150 89480

VIF 6.0 6.0

P 0.000

All else being equal, a 100 pound increase in the weight of the car is associated with a 1.54 increase in horsepower of the auto. All else being equal, a 10 cubic inch increase in the displacement of the engine is associated with a 1.57 increase in the horsepower of the auto. b. Horsepower as a function of weight, displacement, number of cylinders Regression Analysis: horspwr versus weight, displace, cylinder The regression equation is horspwr = 16.7 + 0.0163 weight + 0.105 displace + 2.57 cylinder 151 cases used 4 cases contain missing values Predictor Coef SE Coef T P VIF Constant 16.703 9.449 1.77 0.079 weight 0.016261 0.004592 3.54 0.001 6.2 displace 0.10527 0.05859 1.80 0.074 14.8 cylinder 2.574 2.258 1.14 0.256 7.8 S = 13.63 R-Sq = 69.5% R-Sq(adj) = 68.9% Analysis of Variance Source DF SS MS F Regression 3 62170 20723 111.55 Residual Error 147 27310 186 Total 150 89480

P 0.000

All else being equal, a 100 pound increase in the weight of the car is associated with a 1.63 increase in horsepower of the auto. All else being equal, a 10 cubic inch increase in the displacement of the engine is associated with a 1.05 increase in the horsepower of the auto. All else being equal, one additional cylinder in the engine is associated with a 2.57 increase in the horsepower of the auto.

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Chapter 12: Multiple Regression

12-9

Note that adding the independent variable number of cylinders has not added to the explanatory power of the model. R square has increased marginally. Engine displacement is no longer significant at the .05 level (p-value of .074) and the estimated regression slope coefficient on the number of cylinders is not significantly different from zero. This is due to the strong correlation that exists between cubic inches of engine displacement and the number of cylinders. c. Horsepower as a function of weight, displacement and fuel mileage Regression Analysis: horspwr versus weight, displace, milpgal The regression equation is horspwr = 93.6 + 0.00203 weight + 0.165 displace - 1.24 milpgal 150 cases used 5 cases contain missing values Predictor Coef SE Coef T P VIF Constant 93.57 15.33 6.11 0.000 weight 0.002031 0.004879 0.42 0.678 8.3 displace 0.16475 0.03475 4.74 0.000 6.1 milpgal -1.2392 0.2474 -5.01 0.000 3.1 S = 12.55 R-Sq = 74.2% R-Sq(adj) = 73.6% Analysis of Variance Source DF SS MS F Regression 3 66042 22014 139.77 Residual Error 146 22994 157 Total 149 89036

P 0.000

All else being equal, a 100 pound increase in the weight of the car is associated with a . 203 increase in horsepower of the auto. All else being equal, a 10 cubic inch increase in the displacement of the engine is associated with a 1.6475 increase in the horsepower of the auto. All else being equal, an increase in the fuel mileage of the vehicle by 1 mile per gallon is associated with a reduction in horsepower of 1.2392. Note that the negative coefficient on fuel mileage indicates the trade-off that is expected between horsepower and fuel mileage. The displacement variable is significantly positive, as expected, however, the weight variable is no longer significant. Again, one would expect high correlation among the independent variables. d. Horsepower as a function of weight, displacement, mpg and price Regression Analysis: horspwr versus weight, displace, milpgal, price The regression equation is horspwr = 98.1 - 0.00032 weight + 0.175 displace - 1.32 milpgal +0.000138 price 150 cases used 5 cases contain missing values Predictor Coef SE Coef T P VIF Constant 98.14 16.05 6.11 0.000 weight -0.000324 0.005462 -0.06 0.953 10.3 displace 0.17533 0.03647 4.81 0.000 6.8 milpgal -1.3194 0.2613 -5.05 0.000 3.5 price 0.0001379 0.0001438 0.96 0.339 1.3 S = 12.55 R-Sq = 74.3% R-Sq(adj) = 73.6% Analysis of Variance Source DF SS MS F Regression 4 66187 16547 105.00 Residual Error 145 22849 158 Total 149 89036

P 0.000

12-10

Statistics for Business & Economics, 8th edition

All else being equal, a 100 pound increase in the weight of the car is associated with a reduction of .00324 in horsepower of the auto. All else being equal, a 10 cubic inch increase in the displacement of the engine is associated with a 1.7533 increase in the horsepower of the auto. All else being equal, an increase in the fuel mileage of the vehicle by 1 mile per gallon is associated with a reduction in ho...