Maintenance Refer to the Buena School District bus data. First, add a variable to change the type of bus (diesel or gasoline) to a qualitative variable. If the bus type is diesel, then set the qualitative variable to 0. If the bus type is gasoline, then set the qualitative variable to 1. Develop a regression equation using statistical software with maintenance as the dependent variable and age, miles, and bus type as the independent variables. a. Write out the multiple regression equation analysis. Discuss each of the variables. Maintenance = 102 +5.94 Age + 0.374 Miles - 11.8 Gasoline Indicator Each of the additional years ads $5.94 to the upkeep of cost. Every extra mile adds $0.374 to the maintenance total. Gasoline buses area chaper to maintain than diesel buses by $11.80 per year. b. Determine the value of R2. Interpret. R2 = 65,135/227,692 = 0.286 or 28.6% ROUNDED = 29%. 29% of the variation in maintenance cost is explained by these variables. c. Develop a correlation matrix. Which independent variables have strong or weak correlations with the dependent variable? Do you see any problems with multicollinearity? Age and miles have moderately strong correlations with maintenance cost, however the highest correlation between the variables is between Miles and age with the 0.522 correlation. 0.522 is smaller than 0.70 so there is a good chance that multicollinearity will not be a problem. d. Conduct the global test on the set of independent variables. Interpret. P value is zero, reject null. e. Conduct a test of hypothesis on each of the independent variables. Would you consider deleting any of the variables? If so, which ones? The p-value of the gasoline indicator is the only one of the independent variables that is larger that 0.10. This would be a good variable to consider deleting. f. Rerun the analysis until only significant regression coefficients remain in the analysis. Identify these variables. Maintenance = 106 + 6.17 Age + 0.363 Miles g. Develop a histogram or a stem-and-leaf display of the residuals from the final regression equation developed in part (f). Is it reasonable to conclude that the normality assumption has been met? h. Plot the residuals against the fitted values from the final regression equation developed in part (f) against the fitted values of Y. Plot the residuals on the vertical axis and the fit-ted values on the horizontal axis. 329 329 337 355 357 359 369 380 381 382 390 390 392 392 396 403 406 410 411 414 422 423 424 426 427 428 432 432 433 433 436 439 441 442 444 448 449 450 452 455 457 458 459 459 h. Plot the residuals against the fitted values from the final regression equation developed in part (f) against the fitted values of Y. Plot the residuals on the vertical axis and the fit-ted values on the horizontal axis. 461 462 466 467 468 469 469 471 474 474 475 476 477 478 478 489 492 493 493 494 496 497 501 503 503 504 504 505 514 515 529 540 546 558 561 570 Age Miles 7 3 6 3 8 7 5 9 9 3 2 5 5 8 6 4 3 7 6 4 8 10 4 4 5 7 6 6 9 7 2 9 1 9 2 8 4 6 9 7 2 4 8 11 853 741 819 806 760 751 842 803 882 818 792 799 774 851 784 806 798 866 804 864 869 835 827 757 780 842 819 837 848 817 785 832 823 809 757 790 817 856 831 828 815 817 826 859 y y - y^ 458.829 393.493 440.317 417.088 431.24 421.803 442.496 453.019 481.696 421.444 405.836 426.887 417.812 464.273 427.612 423.258 414.184 463.548 434.872 444.312 470.807 470.805 430.881 405.471 419.99 454.836 440.317 446.851 469.354 445.761 403.295 463.546 410.919 455.197 393.131 442.13 427.251 453.748 463.183 449.754 414.185 427.251 455.198 485.687 Type 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 Diesel Diesel Gasoline Gasoline Diesel Gasoline Gasoline Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Gasoline Diesel Gasoline Gasoline Gasoline Diesel Diesel Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Gasoline Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Diesel Gasoline Diesel 6 6 10 7 4 8 9 9 9 10 9 10 2 6 6 9 10 10 6 7 8 10 7 10 8 8 9 10 11 14 4 11 8 10 12 9 849 799 865 827 800 812 775 815 857 845 816 827 802 821 830 858 836 1008 816 815 839 859 874 883 857 866 842 822 980 895 846 847 870 885 838 844 451.207 433.057 481.695 449.391 421.08 450.116 442.855 457.375 472.621 474.435 457.738 467.901 409.466 441.043 444.31 472.984 471.168 533.604 439.228 445.035 459.917 479.517 466.452 488.229 466.451 469.718 467.176 466.086 529.61 517.265 437.778 481.331 471.17 488.955 484.234 467.902 0 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 Diesel Diesel Gasoline Gasoline Diesel Diesel Gasoline Diesel Gasoline Gasoline Gasoline Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Diesel Gasoline Diesel Diesel Gasoline Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Regression Analysis Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.5348 0.2861 0.2579 46.2484 80 Calculations b3 through b0 intercepts b3 through b0 Standard Error R Square, Standard Error F, Residual df Regression SS, Residual SS Err:502 Err:502 Err:502 Err:502 Err:502 Confidence level t Critical Value Half Width b0 Half Width b1 Half Width b2 Half Width b3 Age Miles Bus Type Analysis of Variance Source Regression Residual Error Total Predictor Constant Age Miles Gasoline Indicator Regression Analysis Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 95% 1.9917 224.8299 4.4363 0.2887 21.8846 Maintenance 0.465 0.450 -0.118 DF Age 1 0.522 -0.068 SS 3 76 79 Coef 1 0.025 MS 65135 162558 227692 SE Coef 102.3 5.939 0.374 -11.8 Miles F 21712 2139 T 112.9 2.227 0.145 10.99 P 10.15 P 0.91 2.67 2.58 -1.07 0.368 0.009 0.012 0.286 0 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.5246 0.2752 0.2564 46.2947 80 Calculations b2, b1, b0 intercepts b2, b1, b0 Standard Error R Square, Standard Error F, Residual df Regression SS, Residual SS Confidence level t Critical Value Half Width b0 Half Width b1 Half Width b2 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 95% 1.9913 224.9068 4.4198 0.2882 Err:502 Err:502 Err:502 Err:502 Err:502 Age 16 14 12 10 Age 8 6 4 2 0 300 350 400 450 500 Maintenance Cost Miles 1200 1000 800 Miles 600 400 200 0 300 350 400 450 Maintenance Cost 500 550 ge 450 500 550 600 ntenance Cost Miles 450 aintenance Cost 500 550 600 Data Set 3 --Buena School District Bus Data Bus Number X1 Maintenance X2 Age X3 Miles X4 Type X5 Bus-Mfg X6 Passenger X7 135 120 200 40 427 759 10 880 481 387 326 861 122 156 887 686 490 370 464 875 883 57 482 704 989 731 75 162 732 751 600 948 358 833 692 61 9 314 329 503 505 466 359 546 427 474 382 422 433 474 558 561 357 329 497 459 355 489 436 455 514 503 380 432 478 406 471 444 493 452 461 496 469 442 414 459 7 10 10 10 7 8 5 9 3 8 9 10 10 12 8 3 10 8 3 9 2 7 11 8 9 6 6 3 9 2 10 9 6 8 8 9 4 11 853 883 822 865 751 870 780 857 818 869 848 845 885 838 760 741 859 826 806 858 785 828 980 857 803 819 821 798 815 757 1008 831 849 839 812 809 864 859 Diesel Diesel Diesel Gasoline Gasoline Diesel Gasoline Gasoline Gasoline Gasoline Diesel Gasoline Gasoline Diesel Diesel Diesel Gasoline Gasoline Gasoline Diesel Gasoline Diesel Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Gasoline Diesel Bluebird Keiser Bluebird Bluebird Keiser Keiser Keiser Keiser Keiser Bluebird Bluebird Bluebird Bluebird Thompson Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Bluebird Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Keiser Keiser Keiser Bluebird Keiser Bluebird Thompson Bluebird Keiser Keiser Thompson 55 Passenger 42 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 14 Passenger 55 Passenger 6 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 6 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 42 Passenger 55 Passenger 55 Passenger 42 Passenger 14 Passenger 55 Passenger 42 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 6 Passenger 396 365 398 43 500 279 693 884 977 38 725 982 724 603 168 45 754 39 671 418 984 953 507 540 695 193 321 918 101 714 678 768 29 554 767 699 954 705 660 520 814 353 457 462 570 439 369 390 469 381 501 432 392 441 448 468 467 478 515 411 504 504 392 423 410 529 477 540 450 390 424 433 428 494 396 458 493 475 476 403 337 492 426 449 2 6 9 9 5 2 9 9 7 6 5 1 8 4 7 6 14 6 8 9 8 10 7 4 2 11 6 5 4 7 7 7 6 4 6 9 10 4 6 10 4 4 815 799 844 832 842 792 775 882 874 837 774 823 790 800 827 830 895 804 866 842 851 835 866 846 802 847 856 799 827 817 842 815 784 817 816 816 827 806 819 836 757 817 Diesel Diesel Diesel Gasoline Gasoline Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Gasoline Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Gasoline Thompson Keiser Thompson Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Keiser Bluebird Bluebird Keiser Keiser Thompson Keiser Keiser Bluebird Thompson Bluebird Bluebird Bluebird Bluebird Bluebird Bluebird Thompson Bluebird Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Keiser Bluebird Bluebird Bluebird Keiser 55 Passenger 55 Passenger 14 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 14 Passenger 55 Passenger 55 Passenger 42 Passenger 14 Passenger 55 Passenger 55 Passenger 14 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 6 Passenger 55 Passenger 55 Passenger 42 Passenger 55 Passenger 42 Passenger 55 Passenger 14 Passenger 55 Passenger 55 Passenger 42 Passenger 42 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger Maintenance Refer to the Buena School District bus data. First, add a variable to change the type of bus (diesel or gasoline) to a qualitative variable. If the bus type is diesel, then set the qualitative variable to 0. If the bus type is gasoline, then set the qualitative variable25to 1. Develop a regression equation using statistical software with maintenance as the dependent variable and age, miles, and bus type as the 20 independent variables. Histogram 15 a. Write out the multiple regression equation analysis. Discuss each of the 10 variables. 5 Frequency = 102 +5.94 Age + 0.374 Miles - 11.8 Gasoline Indicator Maintenance 0 Each of the additional years ads $5.94 to the upkeep of cost. Every extra mile adds $0.374 to the maintenance total. Gasoline buses area chaper to maintain than diesel buses by $11.80 per year. b. Determine the value of R2. Interpret. Bin R2 = 65,135/227,692 = 0.286 or 28.6% ROUNDED = 29%. 29% of the variation in maintenance cost is explained by these variables. c. Develop a correlation matrix. Which independent variables have strong or weak correlations with the dependent variable? Do you see any problems with multicollinearity? Age and miles have moderately strong correlations with maintenance cost, however the highest correlation between the variables is between Miles and age with the 0.522 correlation. 0.522 is smaller than 0.70 so there is a good chance that multicollinearity will not be a problem. d. Conduct the global test on the set of independent variables. Interpret. P value is zero, reject null. e. Conduct a test of hypothesis on each of the independent variables. Would you consider deleting any of the variables? If so, which ones? The p-value of the gasoline indicator is the only one of the independent variables that is larger that 0.10. This would be a good variable to consider deleting. f. Rerun the analysis until only significant regression coefficients remain in the analysis. Identify these variables. Maintenance = 106 + 6.17 Age + 0.363 Miles g. Develop a histogram or a stem-and-leaf display of the residuals from the final regression equation developed in part (f). Is it reasonable to conclude that the normality assumption has been met? The histogram is given in the solution tab. As the histogram is roughly symmetric so normality assmptions is satisfied. h. Plot the residuals against the fitted values from the final regression equation developed in part (f) against the fitted values of Y. Plot the residuals on the vertical axis and the fitted values on the horizontal axis. The graph is given in the solution tab. 329 329 337 355 357 359 369 380 381 382 390 390 392 392 396 403 406 410 411 414 422 423 424 426 427 428 432 432 433 433 436 439 441 442 444 448 449 450 452 455 457 458 459 459 developed in part (f) against the fitted values of Y. Plot the residuals on the vertical axis and the fitted values on the horizontal axis. The graph is given in the solution tab. 461 462 466 467 468 469 469 471 474 474 475 476 477 478 478 489 492 493 493 494 496 497 501 503 503 504 504 505 514 515 529 540 546 558 561 570 Age Miles 7 3 6 3 8 7 5 9 9 3 2 5 5 8 6 4 3 7 6 4 8 10 4 4 5 7 6 6 9 7 2 9 1 9 2 8 4 6 9 7 2 4 8 11 853 741 819 806 760 751 842 803 882 818 792 799 774 851 784 806 798 866 804 864 869 835 827 757 780 842 819 837 848 817 785 832 823 809 757 790 817 856 831 828 815 817 826 859 y y - y^ 458.829 393.493 440.317 417.088 431.24 421.803 442.496 453.019 481.696 421.444 405.836 426.887 417.812 464.273 427.612 423.258 414.184 463.548 434.872 444.312 470.807 470.805 430.881 405.471 419.99 454.836 440.317 446.851 469.354 445.761 403.295 463.546 410.919 455.197 393.131 442.13 427.251 453.748 463.183 449.754 414.185 427.251 455.198 485.687 Type 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 Diesel Diesel Gasoline Gasoline Diesel Gasoline Gasoline Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Gasoline Diesel Gasoline Gasoline Gasoline Diesel Diesel Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Gasoline Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Diesel Gasoline Diesel 6 6 10 7 4 8 9 9 9 10 9 10 2 6 6 9 10 10 6 7 8 10 7 10 8 8 9 10 11 14 4 11 8 10 12 9 849 799 865 827 800 812 775 815 857 845 816 827 802 821 830 858 836 1008 816 815 839 859 874 883 857 866 842 822 980 895 846 847 870 885 838 844 451.207 433.057 481.695 449.391 421.08 450.116 442.855 457.375 472.621 474.435 457.738 467.901 409.466 441.043 444.31 472.984 471.168 533.604 439.228 445.035 459.917 479.517 466.452 488.229 466.451 469.718 467.176 466.086 529.61 517.265 437.778 481.331 471.17 488.955 484.234 467.902 0 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 Diesel Diesel Gasoline Gasoline Diesel Diesel Gasoline Diesel Gasoline Gasoline Gasoline Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Diesel Gasoline Diesel Diesel Gasoline Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Regression Analysis Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.5348 0.2861 0.2579 46.2484 80 Calculations b3 through b0 intercepts b3 through b0 Standard Error R Square, Standard Error F, Residual df Regression SS, Residual SS Err:502 Err:502 Err:502 Err:502 Err:502 Confidence level t Critical Value Half Width b0 Half Width b1 Half Width b2 Half Width b3 Age Miles Bus Type Analysis of Variance Source Regression Residual Error Total Predictor Constant Age Miles Gasoline Indicator Regression Analysis Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 95% 1.9917 224.8299 4.4363 0.2887 21.8846 Maintenance 0.465 0.450 -0.118 DF Age 1 0.522 -0.068 SS 3 76 79 Coef 1 0.025 MS 65135 162558 227692 SE Coef 102.3 5.939 0.374 -11.8 Miles F 21712 2139 T 112.9 2.227 0.145 10.99 P 10.15 P 0.91 2.67 2.58 -1.07 0.368 0.009 0.012 0.286 0 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.5246 0.2752 0.2564 46.2947 80 Calculations b2, b1, b0 intercepts b2, b1, b0 Standard Error R Square, Standard Error F, Residual df Regression SS, Residual SS Confidence level t Critical Value Half Width b0 Half Width b1 Half Width b2 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 95% 1.9913 224.9068 4.4198 0.2882 Err:502 Err:502 Err:502 Err:502 Err:502 Age 16 14 12 10 Age 8 6 4 2 0 300 350 400 450 500 Maintenance Cost Miles 1200 1000 800 Miles 600 400 200 0 300 350 400 450 Maintenance Cost 500 550 ge 450 500 550 600 ntenance Cost Miles 450 aintenance Cost 500 550 600 Data Set 3 --Buena School District Bus Data Bus Number X1 Maintenance X2 Age X3 Miles X4 Type X5 Bus-Mfg X6 Passenger X7 135 120 200 40 427 759 10 880 481 387 326 861 122 156 887 686 490 370 464 875 883 57 482 704 989 731 75 162 732 751 600 948 358 833 692 61 9 314 329 503 505 466 359 546 427 474 382 422 433 474 558 561 357 329 497 459 355 489 436 455 514 503 380 432 478 406 471 444 493 452 461 496 469 442 414 459 7 10 10 10 7 8 5 9 3 8 9 10 10 12 8 3 10 8 3 9 2 7 11 8 9 6 6 3 9 2 10 9 6 8 8 9 4 11 853 883 822 865 751 870 780 857 818 869 848 845 885 838 760 741 859 826 806 858 785 828 980 857 803 819 821 798 815 757 1008 831 849 839 812 809 864 859 Diesel Diesel Diesel Gasoline Gasoline Diesel Gasoline Gasoline Gasoline Gasoline Diesel Gasoline Gasoline Diesel Diesel Diesel Gasoline Gasoline Gasoline Diesel Gasoline Diesel Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Gasoline Diesel Bluebird Keiser Bluebird Bluebird Keiser Keiser Keiser Keiser Keiser Bluebird Bluebird Bluebird Bluebird Thompson Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Bluebird Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Keiser Keiser Keiser Bluebird Keiser Bluebird Thompson Bluebird Keiser Keiser Thompson 55 Passenger 42 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 14 Passenger 55 Passenger 6 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 6 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 42 Passenger 55 Passenger 55 Passenger 42 Passenger 14 Passenger 55 Passenger 42 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 6 Passenger 396 365 398 43 500 279 693 884 977 38 725 982 724 603 168 45 754 39 671 418 984 953 507 540 695 193 321 918 101 714 678 768 29 554 767 699 954 705 660 520 814 353 457 462 570 439 369 390 469 381 501 432 392 441 448 468 467 478 515 411 504 504 392 423 410 529 477 540 450 390 424 433 428 494 396 458 493 475 476 403 337 492 426 449 2 6 9 9 5 2 9 9 7 6 5 1 8 4 7 6 14 6 8 9 8 10 7 4 2 11 6 5 4 7 7 7 6 4 6 9 10 4 6 10 4 4 815 799 844 832 842 792 775 882 874 837 774 823 790 800 827 830 895 804 866 842 851 835 866 846 802 847 856 799 827 817 842 815 784 817 816 816 827 806 819 836 757 817 Diesel Diesel Diesel Gasoline Gasoline Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Gasoline Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Gasoline Thompson Keiser Thompson Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Keiser Bluebird Bluebird Keiser Keiser Thompson Keiser Keiser Bluebird Thompson Bluebird Bluebird Bluebird Bluebird Bluebird Bluebird Thompson Bluebird Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Keiser Bluebird Bluebird Bluebird Keiser 55 Passenger 55 Passenger 14 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 14 Passenger 55 Passenger 55 Passenger 42 Passenger 14 Passenger 55 Passenger 55 Passenger 14 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 6 Passenger 55 Passenger 55 Passenger 42 Passenger 55 Passenger 42 Passenger 55 Passenger 14 Passenger 55 Passenger 55 Passenger 42 Passenger 42 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger Maintenance Age Miles Type 329 329 337 355 357 359 369 380 381 382 390 390 392 392 396 403 406 410 411 414 422 423 424 7 3 6 3 8 7 5 9 9 3 2 5 5 8 6 4 3 7 6 4 8 10 4 853 741 819 806 760 751 842 803 882 818 792 799 774 851 784 806 798 866 804 864 869 835 827 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 1 1 0 0 426 427 428 432 432 433 433 436 439 441 442 444 448 449 450 452 455 457 458 459 459 461 462 4 5 7 6 6 9 7 2 9 1 9 2 8 4 6 9 7 2 4 8 11 6 6 757 780 842 819 837 848 817 785 832 823 809 757 790 817 856 831 828 815 817 826 859 849 799 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 Full model SUMMARY OUTPUT Regression Statistics Multiple R 0.534849 R Square 0.286064 Adjusted R 0.257882 Standard E 46.24844 Observatio 80 ANOVA df Regression Residual Total SS MS 3 65134.59 21711.53 76 162557.8 2138.918 79 227692.4 Coefficients Standard Error t Stat Intercept 102.275 112.885 0.906011 Age 5.938531 2.227408 2.666117 Miles 0.373957 0.144973 2.5795 Type -11.80361 10.98805 -1.074222 Significant variable model SUMMARY OUTPUT Regression Statistics Multiple R 0.524618 R Square 0.275224 Adjusted R 0.256399 Standard E 46.29465 Observatio 80 ANOVA df Regression Residual Total Intercept Age Miles SS MS 2 62666.38 31333.19 77 165026 2143.195 79 227692.4 Coefficients Standard Error t Stat 105.8991 112.9473 0.937598 6.165378 2.219591 2.777709 0.362879 0.14475 2.506939 466 467 468 469 469 471 474 474 475 476 477 478 478 489 492 493 493 494 496 497 501 503 503 504 504 505 514 515 529 540 546 558 561 570 10 7 4 8 9 9 9 10 9 10 2 6 6 9 10 10 6 7 8 10 7 10 8 8 9 10 11 14 4 11 8 10 12 9 865 827 800 812 775 815 857 845 816 827 802 821 830 858 836 1008 816 815 839 859 874 883 857 866 842 822 980 895 846 847 870 885 838 844 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 RESIDUAL OUTPUT Observation Predicted Maintenance Residuals 1 458.5929 -129.5929 2 393.2889 -64.28889 3 440.0896 -103.0896 4 416.8761 -61.87606 5 431.0105 -74.01049 6 421.5792 -62.5792 7 442.2705 -73.27047 8 452.7797 -72.77969 9 481.4472 -100.4472 10 421.2306 -39.23061 11 405.6304 -15.63037 12 426.6667 -36.66666 13 417.5947 -25.59467 14 464.0325 -72.03252 15 427.3888 -31.38884 16 423.0414 -20.04144 17 413.973 -7.973021 18 463.3103 -53.31034 19 434.6464 -23.64643 20 444.0884 -30.08844 21 470.5644 -48.56435 22 470.5572 -47.55721 23 430.6619 -6.661903 24 405.2603 20.73966 25 419.7719 7.228052 26 454.6012 -26.60123 27 440.0896 -8.089624 28 446.6215 -14.62145 29 469.1093 -36.10926 30 445.5292 -12.52924 31 403.0902 32.90979 32 463.3032 -24.30319 33 410.7143 30.28575 34 454.957 -12.95696 35 392.9296 51.07041 36 441.8969 6.103123 37 427.0331 21.96689 38 453.5162 -3.516163 39 462.9403 -10.94031 40 449.5209 5.479083 41 413.9766 43.02341 42 427.0331 30.96689 43 454.9605 4.039464 44 485.4317 -26.43169 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 450.976 10.02399 432.832 29.16796 481.4436 -15.44359 449.158 17.84196 420.8642 47.13584 449.8802 19.11978 442.6191 26.38094 457.1342 13.86576 472.3752 1.624823 474.186 -0.186002 457.4971 17.50288 467.6542 8.345828 409.2592 67.74084 440.8154 37.18462 444.0813 33.9187 472.7381 16.26194 470.9201 21.07991 533.3354 -40.33535 439.001 53.99901 444.8035 49.19652 459.678 36.32203 479.2663 17.73369 466.2134 34.78663 487.9754 15.02458 466.2098 36.7902 469.4757 34.52429 466.932 37.06801 465.8398 39.16022 529.3401 -15.3401 516.9915 -1.991487 437.5566 91.44339 481.0771 58.92286 470.9272 75.07277 488.7012 69.29882 483.9766 77.0234 467.6577 102.3423 F Significance F 10.15071 1.06E-005 P-value Lower 95%Upper 95%Lower 95.0% Upper 95.0% 0.36779 -122.5549 327.1049 -122.5549 327.1049 0.00937 1.502263 10.3748 1.502263 10.3748 0.01182 0.085219 0.662695 0.085219 0.662695 0.28612 -33.68822 10.081 -33.68822 10.081 F Significance F 14.61985 4.15E-006 P-value Lower 95%Upper 95%Lower 95.0% Upper 95.0% 0.351383 -119.0077 330.8059 -119.0077 330.8059 0.006873 1.745608 10.58515 1.745608 10.58515 0.014285 0.074645 0.651113 0.074645 0.651113 Bin Frequency -129.5929 1 -100.601 1 -71.60911 5 -42.61722 6 -13.62532 16 15.36657 18 44.35847 22 73.35036 7 More 4 Histogram of residuals 25 20 15 10 5 Frequency 0 Bin Residuals vs fitted values 150 100 50 Residuals 0 350 370 390 410 430 450 -50 -100 -150 Fitted values 470 490 510 530 550 s 510 530 550 Maintenance Refer to the Buena School District bus data. First, add a variable to change the type of bus (diesel or gasoline) to a qualitative variable. If the bus type is diesel, then set the qualitative variable to 0. If the bus type is gasoline, then set the qualitative variable25to 1. Develop a regression equation using statistical software with maintenance as the dependent variable and age, miles, and bus type as the 20 independent variables. Histogram 15 a. Write out the multiple regression equation analysis. Discuss each of the 10 variables. 5 Frequency = 102 +5.94 Age + 0.374 Miles - 11.8 Gasoline Indicator Maintenance 0 Each of the additional years ads $5.94 to the upkeep of cost. Every extra mile adds $0.374 to the maintenance total. Gasoline buses area chaper to maintain than diesel buses by $11.80 per year. b. Determine the value of R2. Interpret. Bin R2 = 65,135/227,692 = 0.286 or 28.6% ROUNDED = 29%. 29% of the variation in maintenance cost is explained by these variables. c. Develop a correlation matrix. Which independent variables have strong or weak correlations with the dependent variable? Do you see any problems with multicollinearity? Age and miles have moderately strong correlations with maintenance cost, however the highest correlation between the variables is between Miles and age with the 0.522 correlation. 0.522 is smaller than 0.70 so there is a good chance that multicollinearity will not be a problem. d. Conduct the global test on the set of independent variables. Interpret. P value is zero, reject null. e. Conduct a test of hypothesis on each of the independent variables. Would you consider deleting any of the variables? If so, which ones? The p-value of the gasoline indicator is the only one of the independent variables that is larger that 0.10. This would be a good variable to consider deleting. f. Rerun the analysis until only significant regression coefficients remain in the analysis. Identify these variables. Maintenance = 106 + 6.17 Age + 0.363 Miles g. Develop a histogram or a stem-and-leaf display of the residuals from the final regression equation developed in part (f). Is it reasonable to conclude that the normality assumption has been met? The histogram is given in the solution tab. As the histogram is roughly symmetric so normality assmptions is satisfied. h. Plot the residuals against the fitted values from the final regression equation developed in part (f) against the fitted values of Y. Plot the residuals on the vertical axis and the fitted values on the horizontal axis. The graph is given in the solution tab. 329 329 337 355 357 359 369 380 381 382 390 390 392 392 396 403 406 410 411 414 422 423 424 426 427 428 432 432 433 433 436 439 441 442 444 448 449 450 452 455 457 458 459 459 developed in part (f) against the fitted values of Y. Plot the residuals on the vertical axis and the fitted values on the horizontal axis. The graph is given in the solution tab. 461 462 466 467 468 469 469 471 474 474 475 476 477 478 478 489 492 493 493 494 496 497 501 503 503 504 504 505 514 515 529 540 546 558 561 570 Age Miles 7 3 6 3 8 7 5 9 9 3 2 5 5 8 6 4 3 7 6 4 8 10 4 4 5 7 6 6 9 7 2 9 1 9 2 8 4 6 9 7 2 4 8 11 853 741 819 806 760 751 842 803 882 818 792 799 774 851 784 806 798 866 804 864 869 835 827 757 780 842 819 837 848 817 785 832 823 809 757 790 817 856 831 828 815 817 826 859 y y - y^ 458.829 393.493 440.317 417.088 431.24 421.803 442.496 453.019 481.696 421.444 405.836 426.887 417.812 464.273 427.612 423.258 414.184 463.548 434.872 444.312 470.807 470.805 430.881 405.471 419.99 454.836 440.317 446.851 469.354 445.761 403.295 463.546 410.919 455.197 393.131 442.13 427.251 453.748 463.183 449.754 414.185 427.251 455.198 485.687 Type 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 Diesel Diesel Gasoline Gasoline Diesel Gasoline Gasoline Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Gasoline Diesel Gasoline Gasoline Gasoline Diesel Diesel Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Gasoline Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Diesel Gasoline Diesel 6 6 10 7 4 8 9 9 9 10 9 10 2 6 6 9 10 10 6 7 8 10 7 10 8 8 9 10 11 14 4 11 8 10 12 9 849 799 865 827 800 812 775 815 857 845 816 827 802 821 830 858 836 1008 816 815 839 859 874 883 857 866 842 822 980 895 846 847 870 885 838 844 451.207 433.057 481.695 449.391 421.08 450.116 442.855 457.375 472.621 474.435 457.738 467.901 409.466 441.043 444.31 472.984 471.168 533.604 439.228 445.035 459.917 479.517 466.452 488.229 466.451 469.718 467.176 466.086 529.61 517.265 437.778 481.331 471.17 488.955 484.234 467.902 0 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 Diesel Diesel Gasoline Gasoline Diesel Diesel Gasoline Diesel Gasoline Gasoline Gasoline Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Diesel Gasoline Diesel Diesel Gasoline Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Regression Analysis Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.5348 0.2861 0.2579 46.2484 80 Calculations b3 through b0 intercepts b3 through b0 Standard Error R Square, Standard Error F, Residual df Regression SS, Residual SS Err:502 Err:502 Err:502 Err:502 Err:502 Confidence level t Critical Value Half Width b0 Half Width b1 Half Width b2 Half Width b3 Age Miles Bus Type Analysis of Variance Source Regression Residual Error Total Predictor Constant Age Miles Gasoline Indicator Regression Analysis Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 95% 1.9917 224.8299 4.4363 0.2887 21.8846 Maintenance 0.465 0.450 -0.118 DF Age 1 0.522 -0.068 SS 3 76 79 Coef 1 0.025 MS 65135 162558 227692 SE Coef 102.3 5.939 0.374 -11.8 Miles F 21712 2139 T 112.9 2.227 0.145 10.99 P 10.15 P 0.91 2.67 2.58 -1.07 0.368 0.009 0.012 0.286 0 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 0.5246 0.2752 0.2564 46.2947 80 Calculations b2, b1, b0 intercepts b2, b1, b0 Standard Error R Square, Standard Error F, Residual df Regression SS, Residual SS Confidence level t Critical Value Half Width b0 Half Width b1 Half Width b2 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 Err:502 95% 1.9913 224.9068 4.4198 0.2882 Err:502 Err:502 Err:502 Err:502 Err:502 Age 16 14 12 10 Age 8 6 4 2 0 300 350 400 450 500 Maintenance Cost Miles 1200 1000 800 Miles 600 400 200 0 300 350 400 450 Maintenance Cost 500 550 ge 450 500 550 600 ntenance Cost Miles 450 aintenance Cost 500 550 600 Data Set 3 --Buena School District Bus Data Bus Number X1 Maintenance X2 Age X3 Miles X4 Type X5 Bus-Mfg X6 Passenger X7 135 120 200 40 427 759 10 880 481 387 326 861 122 156 887 686 490 370 464 875 883 57 482 704 989 731 75 162 732 751 600 948 358 833 692 61 9 314 329 503 505 466 359 546 427 474 382 422 433 474 558 561 357 329 497 459 355 489 436 455 514 503 380 432 478 406 471 444 493 452 461 496 469 442 414 459 7 10 10 10 7 8 5 9 3 8 9 10 10 12 8 3 10 8 3 9 2 7 11 8 9 6 6 3 9 2 10 9 6 8 8 9 4 11 853 883 822 865 751 870 780 857 818 869 848 845 885 838 760 741 859 826 806 858 785 828 980 857 803 819 821 798 815 757 1008 831 849 839 812 809 864 859 Diesel Diesel Diesel Gasoline Gasoline Diesel Gasoline Gasoline Gasoline Gasoline Diesel Gasoline Gasoline Diesel Diesel Diesel Gasoline Gasoline Gasoline Diesel Gasoline Diesel Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Gasoline Diesel Bluebird Keiser Bluebird Bluebird Keiser Keiser Keiser Keiser Keiser Bluebird Bluebird Bluebird Bluebird Thompson Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Bluebird Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Keiser Keiser Keiser Bluebird Keiser Bluebird Thompson Bluebird Keiser Keiser Thompson 55 Passenger 42 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 14 Passenger 55 Passenger 6 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 6 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 42 Passenger 55 Passenger 55 Passenger 42 Passenger 14 Passenger 55 Passenger 42 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 6 Passenger 396 365 398 43 500 279 693 884 977 38 725 982 724 603 168 45 754 39 671 418 984 953 507 540 695 193 321 918 101 714 678 768 29 554 767 699 954 705 660 520 814 353 457 462 570 439 369 390 469 381 501 432 392 441 448 468 467 478 515 411 504 504 392 423 410 529 477 540 450 390 424 433 428 494 396 458 493 475 476 403 337 492 426 449 2 6 9 9 5 2 9 9 7 6 5 1 8 4 7 6 14 6 8 9 8 10 7 4 2 11 6 5 4 7 7 7 6 4 6 9 10 4 6 10 4 4 815 799 844 832 842 792 775 882 874 837 774 823 790 800 827 830 895 804 866 842 851 835 866 846 802 847 856 799 827 817 842 815 784 817 816 816 827 806 819 836 757 817 Diesel Diesel Diesel Gasoline Gasoline Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Gasoline Gasoline Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Diesel Diesel Diesel Diesel Diesel Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Gasoline Diesel Diesel Gasoline Thompson Keiser Thompson Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Keiser Bluebird Bluebird Keiser Keiser Thompson Keiser Keiser Bluebird Thompson Bluebird Bluebird Bluebird Bluebird Bluebird Bluebird Thompson Bluebird Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Bluebird Keiser Bluebird Bluebird Keiser Bluebird Bluebird Bluebird Keiser 55 Passenger 55 Passenger 14 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 14 Passenger 55 Passenger 55 Passenger 42 Passenger 14 Passenger 55 Passenger 55 Passenger 14 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger 6 Passenger 55 Passenger 55 Passenger 42 Passenger 55 Passenger 42 Passenger 55 Passenger 14 Passenger 55 Passenger 55 Passenger 42 Passenger 42 Passenger 55 Passenger 55 Passenger 55 Passenger 55 Passenger Maintenance Age Miles Type 329 329 337 355 357 359 369 380 381 382 390 390 392 392 396 403 406 410 411 414 422 423 424 7 3 6 3 8 7 5 9 9 3 2 5 5 8 6 4 3 7 6 4 8 10 4 853 741 819 806 760 751 842 803 882 818 792 799 774 851 784 806 798 866 804 864 869 835 827 0 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 1 1 1 0 0 426 427 428 432 432 433 433 436 439 441 442 444 448 449 450 452 455 457 458 459 459 461 462 4 5 7 6 6 9 7 2 9 1 9 2 8 4 6 9 7 2 4 8 11 6 6 757 780 842 819 837 848 817 785 832 823 809 757 790 817 856 831 828 815 817 826 859 849 799 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 Full model SUMMARY OUTPUT Regression Statistics Multiple R 0.534849 R Square 0.286064 Adjusted R 0.257882 Standard E 46.24844 Observatio 80 ANOVA df Regression Residual Total SS MS 3 65134.59 21711.53 76 162557.8 2138.918 79 227692.4 Coefficients Standard Error t Stat Intercept 102.275 112.885 0.906011 Age 5.938531 2.227408 2.666117 Miles 0.373957 0.144973 2.5795 Type -11.80361 10.98805 -1.074222 Significant variable model SUMMARY OUTPUT Regression Statistics Multiple R 0.524618 R Square 0.275224 Adjusted R 0.256399 Standard E 46.29465 Observatio 80 ANOVA df Regression Residual Total Intercept Age Miles SS MS 2 62666.38 31333.19 77 165026 2143.195 79 227692.4 Coefficients Standard Error t Stat 105.8991 112.9473 0.937598 6.165378 2.219591 2.777709 0.362879 0.14475 2.506939 466 467 468 469 469 471 474 474 475 476 477 478 478 489 492 493 493 494 496 497 501 503 503 504 504 505 514 515 529 540 546 558 561 570 10 7 4 8 9 9 9 10 9 10 2 6 6 9 10 10 6 7 8 10 7 10 8 8 9 10 11 14 4 11 8 10 12 9 865 827 800 812 775 815 857 845 816 827 802 821 830 858 836 1008 816 815 839 859 874 883 857 866 842 822 980 895 846 847 870 885 838 844 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 0 RESIDUAL OUTPUT Observation Predicted Maintenance Residuals 1 458.5929 -129.5929 2 393.2889 -64.28889 3 440.0896 -103.0896 4 416.8761 -61.87606 5 431.0105 -74.01049 6 421.5792 -62.5792 7 442.2705 -73.27047 8 452.7797 -72.77969 9 481.4472 -100.4472 10 421.2306 -39.23061 11 405.6304 -15.63037 12 426.6667 -36.66666 13 417.5947 -25.59467 14 464.0325 -72.03252 15 427.3888 -31.38884 16 423.0414 -20.04144 17 413.973 -7.973021 18 463.3103 -53.31034 19 434.6464 -23.64643 20 444.0884 -30.08844 21 470.5644 -48.56435 22 470.5572 -47.55721 23 430.6619 -6.661903 24 405.2603 20.73966 25 419.7719 7.228052 26 454.6012 -26.60123 27 440.0896 -8.089624 28 446.6215 -14.62145 29 469.1093 -36.10926 30 445.5292 -12.52924 31 403.0902 32.90979 32 463.3032 -24.30319 33 410.7143 30.28575 34 454.957 -12.95696 35 392.9296 51.07041 36 441.8969 6.103123 37 427.0331 21.96689 38 453.5162 -3.516163 39 462.9403 -10.94031 40 449.5209 5.479083 41 413.9766 43.02341 42 427.0331 30.96689 43 454.9605 4.039464 44 485.4317 -26.43169 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 450.976 10.02399 432.832 29.16796 481.4436 -15.44359 449.158 17.84196 420.8642 47.13584 449.8802 19.11978 442.6191 26.38094 457.1342 13.86576 472.3752 1.624823 474.186 -0.186002 457.4971 17.50288 467.6542 8.345828 409.2592 67.74084 440.8154 37.18462 444.0813 33.9187 472.7381 16.26194 470.9201 21.07991 533.3354 -40.33535 439.001 53.99901 444.8035 49.19652 459.678 36.32203 479.2663 17.73369 466.2134 34.78663 487.9754 15.02458 466.2098 36.7902 469.4757 34.52429 466.932 37.06801 465.8398 39.16022 529.3401 -15.3401 516.9915 -1.991487 437.5566 91.44339 481.0771 58.92286 470.9272 75.07277 488.7012 69.29882 483.9766 77.0234 467.6577 102.3423 F Significance F 10.15071 1.06E-005 P-value Lower 95%Upper 95%Lower 95.0% Upper 95.0% 0.36779 -122.5549 327.1049 -122.5549 327.1049 0.00937 1.502263 10.3748 1.502263 10.3748 0.01182 0.085219 0.662695 0.085219 0.662695 0.28612 -33.68822 10.081 -33.68822 10.081 F Significance F 14.61985 4.15E-006 P-value Lower 95%Upper 95%Lower 95.0% Upper 95.0% 0.351383 -119.0077 330.8059 -119.0077 330.8059 0.006873 1.745608 10.58515 1.745608 10.58515 0.014285 0.074645 0.651113 0.074645 0.651113 Bin Frequency -129.5929 1 -100.601 1 -71.60911 5 -42.61722 6 -13.62532 16 15.36657 18 44.35847 22 73.35036 7 More 4 Histogram of residuals 25 20 15 10 5 Frequency 0 Bin Residuals vs fitted values 150 100 50 Residuals 0 350 370 390 410 430 450 -50 -100 -150 Fitted values 470 490 510 530 550 s 510 530 550