商务与经济统计习题答案（第8版中文版）SBE8-SM14.doc

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1、Simple Linear RegressionChapter 14Simple Linear RegressionLearning Objectives1.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, and the

2、 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 regression anal

3、ysis output.5.Understand the assumptions necessary for 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 the regression assumptions, recognize outliers, and identify influential observations.7.Know ho

4、w 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.Know the definition of the following terms:independent and depe

5、ndent variablesimple linear regressionregression modelregression equation and estimated regression equation scatter diagramcoefficient of determinationstandard error of the estimateconfidence intervalprediction intervalresidual plotstandardized residual plotoutlierinfluential observationleverageSolu

6、tions:1a. 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 straight line that “best” represents the relationship according to t

7、he 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 will determine th

8、e equation of a straight line that “best” represents the relationship according to the least squares criterion.d. Summations needed to compute the slope and y-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 linear relationship between x

9、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.Summations needed to compute the sl

10、ope and y-intercept are:e.pounds5.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 straight line that “best” represents the r

11、elationship 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 linear relationship between x and y.c.Summations

12、 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.e7.a. b.Let x = DJIA and y = S&P. Summations needed to compute the slope and y-intercept are:c. or approximate

13、ly 15008.a.Summations needed to compute the slope and y-intercept are:b.Increasing the number of times an ad is aired by one will increase the number of household exposures by approximately 3.07 million.c.9.a.b.Summations needed to compute the slope and y-intercept are:c.10.a. b.Let x = performance

14、score and y = overall rating. Summations needed to compute the slope and y-intercept are:c. or approximately 8411.a.b.There appears to be a linear relationship between the variables.c.The summations needed to compute the slope and the y-intercept are:d.12.a.b.There appears to be a positive linear re

15、lationship 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 the y-intercept are:c.or approximately $13,080. The agents request for an aud

16、it 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 error and the total sum of squares areThus, SSR = SST - SSE = 80 - 12.4 = 67.6b.r2 = SSR

17、/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 dependent variable are:The sum of squares due to error and the total sum of squares areThus,

18、 SSR = SST - SSE = 114.80 - 6.33 = 108.47b.r2 = SSR/SST = 108.47/114.80 = .945The least squares line provided an excellent fit; 94.5% of the variability in y has been explained by the estimated regression equation.c.Note: the sign for r is negative because the slope of the estimated regression equat

19、ion 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 the total sum of squares areThus, SSR = SST - SSE = 11.2 - 5.3 = 5.9r2 = SSR/SST = 5.9/11.2 = .527We see that 52.7% of the variability in y has been explai

20、ned 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 areThus, SSR = SST - SSE = 335,000 - 85,135.14 = 249,864.86b.r2 = SSR/SST = 249,864.86/335,000 = .746We see that 74.6% of the

21、 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 the total sum of squares areThus, SSR = SST - SSE = 47,582.10 - 7547.14 = 40,034.96b.r2 = SSR/SST = 40,034.96/47,58

22、2.10 = .84We 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 and y-intercept are: b. The sum of squares due to error and the total sum of squares areThus, SSR = SST - SSE = 11,373.09 20

23、17.37 = 9355.72r2 = SSR/SST = 9355.72/11,373.09 = .82We see that 82% of the variability in y has been explained by the least squares line.c. or approximately $173,50021.a.The summations needed in this problem are:b.$7.60c.The sum of squares due to error and the total sum of squares are:Thus, SSR = S

24、ST - SSE = 5,648,333.33 - 233,333.33 = 5,415,000r2 = SSR/SST = 5,415,000/5,648,333.33 = .9587We see that 95.87% of the variability in y has been explained by the estimated 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 squa

25、res are:Thus, SSR = SST - SSE = 1998 - 1272.4495 = 725.5505r2 = SSR/SST = 725.5505/1998 = 0.3631Approximately 37% of the variability in change in executive compensation is explained by the two-year change in the return on equity. c.It reflects a linear relationship that is between weak and strong.23

26、.a.s2 = MSE = SSE / (n - 2) = 12.4 / 3 = 4.133b.c.d.t.025 = 3.182 (3 degrees of freedom)Since t = 4.04 t.05 = 3.182 we reject H0: b1 = 0e.MSR = SSR / 1 = 67.6F = MSR / MSE = 67.6 / 4.133 = 16.36F.05 = 10.13 (1 degree of freedom numerator and 3 denominator)Since F = 16.36 F.05 = 10.13 we reject H0: b

27、1 = 0Source of VariationSum of SquaresDegrees of FreedomMean SquareFRegression67.6167.616.36Error12.434.133Total80.0424.a.s2 = MSE = SSE / (n - 2) = 6.33 / 3 = 2.11b.c.d.t.025 = 3.182 (3 degrees of freedom)Since t = -7.18 F.05 = 10.13 we reject H0: b1 = 0Source of VariationSum of SquaresDegrees of F

28、reedomMean SquareFRegression108.471 108.47 51.41Error 6.333 2.11Total114.80425.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 not appear to be related.c.MSR = SSR/1 = 5.90 /1 = 5.90F = MSR/MSE = 5

29、.90/1.77 = 3.33F.05 = 10.13 (1 degree of freedom numerator and 3 denominator)Since F = 3.33 t.025 = 2.776 we reject H0: b1 = 0b.MSR = SSR / 1 = 249,864.86 / 1 = 249.864.86F = MSR / MSE = 249,864.86 / 21,283.79 = 11.74F.05 = 7.71 (1 degree of freedom numerator and 4 denominator)Since F = 11.74 F.05 =

30、 7.71 we reject H0: b1 = 0c.Source of VariationSum of SquaresDegrees of FreedomMean SquareFRegression249864.861 249864.86 11.74Error85135.14421283.79Total335000527.The sum of squares due to error and the total sum of squares are:SSE = SST = 2442Thus, SSR = SST - SSE = 2442 - 170 = 2272MSR = SSR / 1

31、= 2272SSE = SST - SSR = 2442 - 2272 = 170MSE = SSE / (n - 2) = 170 / 8 = 21.25F = MSR / MSE = 2272 / 21.25 = 106.92F.05 = 5.32 (1 degree of freedom numerator and 8 denominator)Since F = 106.92 F.05 = 5.32 we reject H0: b1 = 0.Years of experience and sales are related.28.SST = 411.73 SSE = 161.37 SSR

32、 = 250.36MSR = SSR / 1 = 250.36MSE = SSE / (n - 2) = 161.37 / 13 = 12.413F = MSR / MSE = 250.36 / 12.413= 20.17F.05 = 4.67 (1 degree of freedom numerator and 13 denominator)Since F = 20.17 F.05 = 4.67 we reject H0: b1 = 0.29.SSE = 233,333.33 SST = 5,648,333.33 SSR = 5,415,000MSE = SSE/(n - 2) = 233,

33、333.33/(6 - 2) = 58,333.33MSR = SSR/1 = 5,415,000F = MSR / MSE = 5,415,000 / 58,333.25 = 92.83 Source of VariationSum of SquaresDegrees of FreedomMean SquareFRegression5,415,000.001 5,415,000 92.83Error 233,333.334 58,333.33Total5,648,333.335F.05 = 7.71 (1 degree of freedom numerator and 4 denominat

34、or)Since F = 92.83 7.71 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 = 45,833.9286t.025 = 2.571Since t = 1.69 F.01 = 8.53 we reject H0: b1 = 0.32.a.s = 2.033b.10.6 3.182 (1.11) = 10.6 3.53or 7

35、.07 to 14.13c.d.10.6 3.182 (2.32) = 10.6 7.38or 3.22 to 17.9833.a.s = 1.453b.24.69 3.182 (.68) = 24.69 2.16or 22.53 to 26.85c.d.24.69 3.182 (1.61) = 24.69 5.12or 19.57 to 29.8134.s = 1.332.28 3.182 (.85) = 2.28 2.70or -.40 to 4.982.28 3.182 (1.58) = 2.28 5.03or -2.27 to 7.3135.a.s = 145.89 2,033.78

36、2.776 (68.54) = 2,033.78 190.27or $1,843.51 to $2,224.05b.2,033.78 2.776 (161.19) = 2,033.78 447.46or $1,586.32 to $2,481.2436.a.b.s = 3.523280.859 2.160 (1.055) = 80.859 2.279or 78.58 to 83.14c.80.859 2.160 (3.678) = 80.859 7.944or 72.92 to 88.8037.a.s2 = 1.88 s = 1.3713.08 2.571 (.52) = 13.08 1.34

37、or 11.74 to 14.42 or $11,740 to $14,420b.sind = 1.4713.08 2.571 (1.47) = 13.08 3.78or 9.30 to 16.86 or $9,300 to $16,860c.Yes, $20,400 is much larger than anticipated.d.Any deductions exceeding the $16,860 upper limit could suggest an audit.38.a.b.s2 = MSE = 58,333.33 s = 241.525046.67 4.604 (267.50) = 5046.67 1231.57or $3815.10 to $6278.24c.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 cause co