1、商务与经济统计习题答案（第 8 版，中文版）SBE8Chapter 14 Simple Linear Regression Learning Objectives 1. Understand how regression analysis can be used to develop an equation that estimates mathematically how two variables are related.2. Understand the differences between the regression model, the regression equation,
2、and the estimated regression equation. 3. Know how to fit an estimated regression equation to a set of sample data based upon the least-squares method. 4. Be able to determine how good a fit is provided by the estimated regression equation and compute the sample correlation coefficient from the regr
3、ession analysisoutput.5.Understandtheassumptionsnecessaryfor statistical inference and be able to test for a significant relationship. 6. Learn how to use a residual plot to make a judgement as to the validity of theregressionassumptions,recognize outliers, and identify influential observations. 7.
4、Know how to develop confidence interval estimates of y given a specific value of x in both the case of a mean value of y and an individual value of y. 8. Be able to compute the sample correlation coefficient from the regression analysis output. 9. Knowthedefinitionofthefollowingterms:independentand
5、dependentvariablesimplelinearregressionregressionmodel regressionequationandestimatedregressionequationscatter diagram coefficient of determination standard error of the estimate confidence interval prediction interval residual plot standardized residual plot outlier influential observation leverage
6、 Solutions: 1 a. b. There appears to be a linear relationship between x and y. c. Many different straight lines can be drawn to provide a linear approximation of the relationship between x and y; in part d we will determine theequation of a straight line that “best” represents the relationship accor
7、ding to the least squares criterion. d. Summations needed to compute the slope and y-intercept are: e. 2. a. b. There appears to be a linear relationship between x and y. c. Many different straight lines can be drawn to provide a linear approximation of the relationship between x and y; in part d we
8、 will determine the equation of a straight line that “best” represents the relationship according to the least squares criterion. d. Summations needed to compute the slope andy-intercept are: e. 3. a. b. Summations needed to compute the slope and y-intercept are: c. 4. a. b. There appears to be a li
9、near relationship between x and y. c. Many different straight lines can be drawn to provide a linear approximation of the relationship between x and y; in part d we will determine the equation of a straight line that “best” represents the relationship according to the least squares criterion. d. Sum
10、mations needed to compute the slope and y-intercept are: e. pounds 5. a. b. There appears to be a linear relationship between x and y. c. Many different straight lines can be drawn to provide a linear approximation of the relationship between x and y; in part d we will determine the equation of a st
11、raight line that “best” representsthe relationship according to the least squares criterion. Summations needed to compute the slope and y-intercept are: d. A one million dollar increase in media expenditures will increase case sales by approximately 14.42 million. e. 6. a. b. There appears to be a l
12、inear relationship between x and y. c. Summations needed to compute the slope and y-intercept are: d. A one percent increase in the percentage of flights arriving on time will decrease the number of complaints per 100,000 passengers by 0.07. e 7. a. b. Let x = DJIA and y = SP. Summations needed to c
13、ompute the slope and y-intercept are: c. or approximately 1500 8. a. Summations needed to compute the slopeand y-intercept are: b. Increasing the number of times an ad is aired byonewillincreasethenumberofhouseholdexposuresby approximately 3.07 million. c. 9. a. b. Summations needed to compute the s
14、lope and y-intercept are: c. 10. a. b. Let x = performance score and y = overall rating. Summations needed to compute the slope and y-intercept are: c. or approximately 84 11. a. b. There appears to be a linear relationship between the variables. c. The summations needed to compute the slope and the
15、 y-intercept are: d. 12. a. b. There appears to be a positive linear relationship between the number of employees and the revenue. c. Let x = number of employees and y = revenue. Summations needed to compute the slope and y-intercept are: d. 13. a. b. The summations needed to compute the slope and t
16、he y-intercept are: c. or approximately $13,080. The agents request for an audit appears to be justified. 14. a. b. The summations needed to compute the slope and the y-intercept are: c. 15. a. The estimated regression equation and the mean for the dependent variable are: The sum of squares due to e
17、rror and the total sum of squares are Thus, SSR = SST - SSE = 80 - 12.4 = 67.6 b. r2 = SSR/SST = 67.6/80 = .845The least squares line provided a very good fit; 84.5% of the variability in y has been explained by the least squares line. c. 16. a. The estimated regression equation and the mean for the
18、 dependent variable are: The sum of squares due to error and the total sum of squares are Thus, SSR = SST - SSE = 114.80 - 6.33 = 108.47 b. r2 = SSR/SST = 108.47/114.80 = .945 The least squares line provided an excellent fit; 94.5% of the variability in y has been explained by the estimated regressi
19、on equation. c. Note: the sign for r is negative because the slope of the estimated regression equation is negative. (b1 = -1.88) 17. The estimated regression equation and the mean for the dependent variable are: The sum of squares due to error and thetotal sum of squares are Thus, SSR = SST - SSE =
20、 11.2 - 5.3 = 5.9 r2 = SSR/SST = 5.9/11.2 = .527 We see that 52.7% of the variability in y has been explained by the least squares line. 18. a. The estimated regression equation and the mean for the dependent variable are: The sum of squares due to error and the total sum of squares are Thus, SSR =
21、SST - SSE = 335,000 - 85,135.14 = 249,864.86 b. r2 = SSR/SST= 249,864.86/335,000 = .746 We see that 74.6% of the variability in y has been explained by the least squares line. c. 19. a. The estimated regression equation and the mean for the dependent variable are: The sum of squares due to error and
22、 the total sum of squares are Thus, SSR = SST - SSE = 47,582.10 - 7547.14 = 40,034.96 b. r2 = SSR/SST =40,034.96/47,582.10 = .84 We see that 84% of the variability in y has been explained by the least squares line. c. 20. a. Let x = income and y = home price. Summations needed to compute the slope a
23、nd y- intercept are: b. The sum of squares due to error and the total sum of squares are Thus, SSR = SST - SSE = 11,373.09 2017.37 = 9355.72r2 = SSR/SST = 9355.72/11,373.09 = .82 We see that 82% of the variability in y has been explained by the least squares line. c. or approximately $173,500 21. a.
24、 The summations needed in this problem are: b. $7.60 c. The sum of squares due to error and the total sum of squares are: Thus, SSR = SST - SSE = 5,648,333.33 - 233,333.33= 5,415,000 r2 = SSR/SST = 5,415,000/5,648,333.33 = .9587 We seethat 95.87% of the variability in y has been explained by the est
25、imated regression equation. d. 22. a. The summations needed in this problem are: b. The sum of squares due to error and the total sum of squares are: Thus, SSR = SST - SSE = 1998 - 1272.4495 = 725.5505 r2 =SSR/SST = 725.5505/1998 = 0.3631 Approximately 37% of the variability in change in executive c
26、ompensation is explained by the two-year change in the return on equity. c. It reflects a linearrelationship that is between weak and strong. 23. a. s2 = MSE = SSE / (n - 2) = 12.4 / 3 = 4.133 b. c. d. t.025 = 3.182 (3 degrees of freedom) Since t = 4.04t.05 = 3.182 we reject H0: b1 = 0 e. MSR = SSR
27、/ 1 = 67.6 F = MSR / MSE = 67.6 / 4.133 = 16.36 F.05 = 10.13 (1 degree offreedom numerator and 3 denominator) Since F = 16.36F.05 =10.13 we reject H0: b1 = 0 Source of Variation Sum of Squares Degrees of Freedom Mean Square F Regression 67.6 1 67.6 16.36 Error 12.4 3 4.133 Total 80.0 4 24. a. s2 = M
28、SE = SSE / (n - 2) = 6.33 / 3 =2.11 b. c. d. t.025 = 3.182 (3 degrees of freedom) Since t = -7.18-t.025 = -3.182 we reject H0: b1 = 0 e. MSR = SSR / 1 = 8.47 F = MSR/ MSE = 108.47 / 2.11 = 51.41 F.05 = 10.13 (1 degree of freedom numerator and 3 denominator) Since F = 51.41 F.05 = 10.13 we reject H0:
29、 b1 = 0 Source of Variation Sum of Squares Degrees of Freedom Mean Square F Regression 108.47 1 108.47 51.41 Error 6.333 2.11 Total 114.80 4 25. a. s2 = MSE = SSE / (n - 2) = 5.30 / 3 = 1.77b. t.025 = 3.182 (3 degrees of freedom) Since t = 1.82 t.025 = 3.182 we cannot reject H0: b1 = 0; x and y do n
30、ot appear to be related. c.MSR = SSR/1 = 5.90 /1 = 5.90 F = MSR/MSE = 5.90/1.77 = 3.33 F.05= 10.13 (1 degree of freedom numerator and 3 denominator) Since F= 3.33F.05 = 10.13 we cannot reject H0: b1 = 0; x and y do not appear to be related. 26. a. s2 = MSE = SSE / (n - 2) = 85,135.14 / 4 = 21,283.79
31、 t.025 = 2.776 (4 degrees of freedom) Since t = 3.43t.025= 2.776 we reject H0: b1 = 0 b. MSR = SSR / 1 = 249,864.86 / 1 = 249.864.86 F = MSR / MSE = 249,864.86 / 21,283.79 = 11.74 F.05 =7.71 (1 degree of freedom numerator and 4 denominator) Since F =11.74F.05 = 7.71 we reject H0: b1 = 0 c. Source of
32、 Variation Sum of Squares Degrees of Freedom Mean Square F Regression *.86 1*.86 11.74 Error *.14 4 *.79 Total * 5 27. The sum of squares due to error and the total sum of squares are: SSE = SST =2442 Thus, SSR = SST - SSE = 2442 - 170 = 2272 MSR = SSR / 1 =2272 SSE = SST - SSR = 2442 - 2272 = 170 M
33、SE = SSE / (n - 2) = 170/ 8 = 21.25 F = MSR / MSE = 2272 / 21.25 = 106.92 F.05 = 5.32 (1degree of freedom numerator and 8 denominator) Since F = 106.92F.05 = 5.32 we reject H0: b1 = 0. Years of experience and sales are related. 28. SST = 411.73 SSE = 161.37 SSR = 250.36 MSR = SSR / 1 = 250.36 MSE =
34、SSE / (n - 2) = 161.37 / 13 = 12.413 F = MSR / MSE =250.36 / 12.413= 20.17 F.05 = 4.67 (1 degree of freedom numerator and 13 denominator) Since F = 20.17F.05 = 4.67 we reject H0: b1 = 0. 29. SSE = 233,333.33 SST = 5,648,333.33 SSR = 5,415,000 MSE = SSE/(n - 2) = 233,333.33/(6 - 2) = 58,333.33 MSR =
35、SSR/1 = 5,415,000 F = MSR / MSE = 5,415,000 / 58,333.25 = 92.83 Source of Variation Sum of Squares Degrees of Freedom Mean Square F Regression 5,415,000.00 1 5,415,000 92.83 Error 233,333.33 4 58,333.33 Total 5,648,333.33 5 F.05 = 7.71 (1 degree of freedom numerator and 4 denominator) Since F = 92.8
36、37.71 we reject H0: b1 = 0. Production volume and total cost are related. 30. Using the computations from Exercise 22, SSE = 1272.4495 SST = 1998 SSR = 725.5505 = 45,833.9286 t.025 = 2.571 Since t = 1.692.571, we cannot reject H0: b1 = 0 There is no evidence of a significant relationship between x a
37、nd y. 31. SST = 11,373.09 SSE = 2017.37 SSR = 9355.72 MSR = SSR / 1 = 9355.72 MSE = SSE / (n - 2) = 2017.37/ 16 = 126.0856 F = MSR / MSE= 9355.72/ 126.0856 = 74.20 F.01 = 8.53 (1 degree of freedom numerator and 16 denominator) Since F = 74.20 F.01 = 8.53 we reject H0: b1 = 0. 32. a. s = 2.033 b. 10.
38、6 3.182 (1.11) = 10.6 3.53or 7.07 to 14.13 c. d. 10.6 3.182 (2.32) = 10.6 7.38 or 3.22 to 17.9833. a. s = 1.453 b. 24.69 3.182 (.68) = 24.69 2.16 or 22.53 to 26.85c. d. 24.69 3.182 (1.61) = 24.69 5.12 or 19.57 to 29.81 34. s = 1.332.28 3.182 (.85) = 2.28 2.70 or -.40 to 4.98 2.28 3.182 (1.58) =2.28
39、5.03 or -2.27 to 7.31 35. a. s = 145.89 2,033.78 2.776 (68.54)= 2,033.78 190.27 or $1,843.51 to $2,224.05 b. 2,033.78 2.776 (161.19) = 2,033.78 447.46 or $1,586.32 to $2,481.24 36. a. b. s = 3.5232 80.859 2.160 (1.055) = 80.859 2.279 or 78 .58 to 83.14 c.80.859 2.160 (3.678) = 80.859 7.944 or 72.92
40、to 88.80 37. a. s2 =1.88 s = 1.37 13.08 2.571 (.52) = 13.08 1.34 or 11.74 to 14.42 or$11,740 to $14,420 b. sind = 1.47 13.08 2.571 (1.47) = 13.08 3.78or 9.30 to 16.86 or $9,300 to $16,860 c. Yes, $20,400 is much larger than anticipated. d. Any deductions exceeding the $16,860 upper limit could sugge
41、st an audit. 38. a. b. s2= MSE = 58,333.33 s = 241.52 5046.67 4.604 (267.50) = 5046.67 1231.57 or $3815.10 to$6278.24 c. Based on one month, $6000 is not out of line since$3815.10 to $6278.24 is the prediction interval. However, a sequence of five to seven months with consistently high costs should
42、cause concern. 39. a. Summations needed to compute the slope and y- intercept are: b. SST = 39,065.14 SSE = 4145.141 SSR = 34,920.000 r2= SSR/SST = 34,920.000/39,065.141 = 0.894 The estimated regression equation explained 89.4% of the variability in y; a very good fit. c. s2 = MSE = 4145.141/8 = 518
43、.143 270.63 2.262 (8.86) = 270.63 20. 04or 250.59 to 290.67 d. 270.63 2.262 (24.42) = 270.63 55.24 or215.39 to 325.87 40. a. 9 b. = 20.0 + 7.21x c. 1.3626 d. SSE = SST - SSR = 51,984.1 - 41,587.3 = 10,396.8 MSE = 10,396.8/7 = 1,485.3 F = MSR / MSE = 41,587.3 /1,485.3 = 28.00 F.05 = 5.59 (1 degree of
44、freedom numerator and 7 denominator) Since F = 28 F.05 = 5.59 we reject H0: B1 = 0. e. = 20.0 + 7.21(50) = 380.5 or $380,500 41. a. =6.1092 + .8951x b. t.025 = 2.306 (1 degree of freedom numerator and 8 denominator) Since t = 6.01t.025 = 2.306 we reject H0: B1 = 0 c.= 6.1092 + .8951(25) = 28.49 or $
45、28.49 per month 42 a. = 80.0 + 50.0x b. 30 c. F = MSR / MSE = 6828.6/82.1 = 83.17 F.05 = 4.20 (1 degreeof freedom numerator and 28 denominator) Since F = 83.17F.05 =4.20 we reject H0: B1 = 0. Branch office sales are related to the salespersons. d. = 80 + 50 (12) = 680 or $680,000 43. a. The Minitab
46、output is shown below: The regression equation is Price = - 11.8 +2.18 Income Predictor Coef SE Coef T P Constant -11.80 12.84 -0.92 0.380 Income 2.1843 0.2780 7.86 0.000 S = 6.634 R-Sq = 86.1% R- Sq(adj) = 84.7% Analysis of Variance Source DF SS MS F P Regression 1 2717.9 2717.9 61.75 0.000 Residua
47、l Error 10 440.1 44.0 Total 113158.0 Predicted Values for New Observations New Obs Fit SE Fit 95.0% CI 95.0% PI 1 75.79 2.47 ( 70.29, 81.28) ( 60.02, 91.56) b. r2 = .861. Theleast squares line provided a very good fit. c. The 95% confidence interval is 70.29 to 81.28 or $70,290 to $81,280. d. The 95
48、% prediction interval is 60.02 to 91.56 or $60,020 to $91,560. 44. a/b. The scatter diagram shows a linear relationship between the two variables. c. The Minitab output is shown below: The regression equation is Rental$ =37.1 - 0.779 Vacancy% Predictor Coef SE Coef T P Constant 37.066 3.530 10.50 0.000 Vacancy% -0.7791 0.2226 -3.50 0.003 S = 4.889 R- Sq = 43.4% R-Sq(adj) = 39.8% Analysis of Variance Source DF SS MS F P Regression 1 292.89 292.89 12.26 0.003 Residual Error 16 382.3723.90 Total 17 675.26 Predicted Values for New Observations New Obs Fit SE Fit 95.0% CI 95.0% PI 1 17.59 2.51 (
