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Population size (Thousands of Residents). Yearly Revenue, Y (Thousands of Dollars) The Tasty Sub Shop Revenue Data Restaurant     Population Size, X1        Business Rating, X2    Yearly Revenue, Y          1                     20.8                                   3                                    527.1 2                     27.5                                   2                                    548.7 3                     32.3                                   6                                    767.2 4                     37.2                                   5                                    722.9 5                     39.6                                   8                                    826.3 6                     45.1                                   3                                    810.5 7                     49.9                                   9                                    1040.5 8                     55.4                                   5                                    1033.6 9                     61.7                                   4                                    1090.3 10                   64.6                                   7                                    1235.8 Questions 2. Using the t statistic and appropriate critical values, test Ho: βj = 0 versus Ha: βj ≠ 0 by setting α equal to .05. Which independent variables are significantly related to y in the model with α = .05?  3. Using the t statistic and appropriate critical values, test Ho: βj = 0 versus Ha: βj ≠ 0 by setting α equal to .01. Which independent variables are significantly related to y in the model with α = .01? 4. Find the p-value for testing Ho: βj = 0 versus Ha: βj ≠ 0 on the output. Using the p-value, determine whether we can reject Ho by setting α equal to .10, .05, .01, and .001. What do you conclude about the significance of the independent variables in the model? 5. Calculate the 95 percent confidence interval for βj. Discuss one practical application of this interval.  6. Calculate the 99 percent confidence interval for βj.

Population size (Thousands of Residents). Yearly Revenue, Y (Thousands of Dollars) The Tasty Sub Shop Revenue Data Restaurant     Population Size, X1        Business Rating, X2    Yearly Revenue, Y          1                     20.8                                   3                                    527.1 2                     27.5                                   2                                    548.7 3                     32.3                                   6                                    767.2 4                     37.2                                   5                                    722.9 5                     39.6                                   8                                    826.3 6                     45.1                                   3                                    810.5 7                     49.9                                   9                                    1040.5 8                     55.4                                   5                                    1033.6 9                     61.7                                   4                                    1090.3 10                   64.6                                   7                                    1235.8 Questions 2. Using the t statistic and appropriate critical values, test Ho: βj = 0 versus Ha: βj ≠ 0 by setting α equal to .05. Which independent variables are significantly related to y in the model with α = .05?  3. Using the t statistic and appropriate critical values, test Ho: βj = 0 versus Ha: βj ≠ 0 by setting α equal to .01. Which independent variables are significantly related to y in the model with α = .01? 4. Find the p-value for testing Ho: βj = 0 versus Ha: βj ≠ 0 on the output. Using the p-value, determine whether we can reject Ho by setting α equal to .10, .05, .01, and .001. What do you conclude about the significance of the independent variables in the model? 5. Calculate the 95 percent confidence interval for βj. Discuss one practical application of this interval.  6. Calculate the 99 percent confidence interval for βj.

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Population size (Thousands of Residents). Yearly Revenue, Y (Thousands of Dollars)

The Tasty Sub Shop Revenue Data
Restaurant     Population Size, X1        Business Rating, X2    Yearly Revenue, Y
        
1                     20.8                                   3                                    527.1
2                     27.5                                   2                                    548.7
3                     32.3                                   6                                    767.2
4                     37.2                                   5                                    722.9
5                     39.6                                   8                                    826.3
6                     45.1                                   3                                    810.5
7                     49.9                                   9                                    1040.5
8                     55.4                                   5                                    1033.6
9                     61.7                                   4                                    1090.3
10                   64.6                                   7                                    1235.8
Questions
2. Using the t statistic and appropriate critical values, test Ho: βj = 0 versus Ha: βj ≠ 0 by setting α equal to .05. Which independent variables are significantly related to y in the model with α = .05? 

3. Using the t statistic and appropriate critical values, test Ho: βj = 0 versus Ha: βj ≠ 0 by setting α equal to .01. Which independent variables are significantly related to y in the model with α = .01?

4. Find the p-value for testing Ho: βj = 0 versus Ha: βj ≠ 0 on the output. Using the p-value, determine whether we can reject Ho by setting α equal to .10, .05, .01, and .001. What do you conclude about the significance of the independent variables in the model?

5. Calculate the 95 percent confidence interval for βj. Discuss one practical application of this interval. 

6. Calculate the 99 percent confidence interval for βj.    

14.5 Testing the Significance of an Independent Variable 543 t Statistics and p-Values for Testing the Significance of the Intercept, x, and x, in the Tasty Sub Shop Revenue Model y = Bo + B,x, + B,X2 + E TABLE 14.7 (a) Calculation of the t statistics Null b; Independent Variable Hypothesis bị Sbj p-Value bo t = 125.29 Intercept Ho: Bo = 0 bo = 125.29 %3D Sbo = 40.93 =D 3.06 .0183 %3D Spo 40.93 b1 14.1996 = 15.6 < .001 b, = 14.1996 = 0.91 %3D X1 Ho: B, = 0 .91 22.811 b2 t = = 3.95 .0055 %D Họ: B2 = 0 b2 = 22.811 Sb, = 5.769 %3D 5.769 %3D X2 (b) The MINITAB output Coef SE Coef T P Predictor 125.29 40.93 3.06 0.018 Constant population 14.1996 bus rating 0.91 15.6 0.000 22.811 5.769 3.95 0.006 (c) The Excel output Lower 95% Upper 95% t Stat P-value Coefficients Standard Error 222.0807 0.0183 28.4969 40.9333 3.06 16.3515 125.289 12.0478 Intercept 15.60 1.07E-06 0.9100 36.4527 14.1996 9.1686 population bus_rating 3.95 0.0055 5.7692 22.8107 C DODODLE 14 3 The Tasty Sub Shop Case: t statistics and related p-values
Transcribed Image Text:14.5 Testing the Significance of an Independent Variable 543 t Statistics and p-Values for Testing the Significance of the Intercept, x, and x, in the Tasty Sub Shop Revenue Model y = Bo + B,x, + B,X2 + E TABLE 14.7 (a) Calculation of the t statistics Null b; Independent Variable Hypothesis bị Sbj p-Value bo t = 125.29 Intercept Ho: Bo = 0 bo = 125.29 %3D Sbo = 40.93 =D 3.06 .0183 %3D Spo 40.93 b1 14.1996 = 15.6 < .001 b, = 14.1996 = 0.91 %3D X1 Ho: B, = 0 .91 22.811 b2 t = = 3.95 .0055 %D Họ: B2 = 0 b2 = 22.811 Sb, = 5.769 %3D 5.769 %3D X2 (b) The MINITAB output Coef SE Coef T P Predictor 125.29 40.93 3.06 0.018 Constant population 14.1996 bus rating 0.91 15.6 0.000 22.811 5.769 3.95 0.006 (c) The Excel output Lower 95% Upper 95% t Stat P-value Coefficients Standard Error 222.0807 0.0183 28.4969 40.9333 3.06 16.3515 125.289 12.0478 Intercept 15.60 1.07E-06 0.9100 36.4527 14.1996 9.1686 population bus_rating 3.95 0.0055 5.7692 22.8107 C DODODLE 14 3 The Tasty Sub Shop Case: t statistics and related p-values
Multiple Regression and Model Building Excel and MINITAB Outputs of a Regression Analysis of the Tasty Sub Shop Revenue Data in Table 14.1 Using the Model y = Bo + B,x, + Bx2 + E FIGURE 14.4 %3D (a) The Excel output Regression Statistics Multiple R 0.9905 R Square 0.9810 8 Adjusted R Square 0.9756 9 36.6856 7 Standard Error Observations (35) ANOVA df SS 486355.7 10 MS F Significance F Regression 243177.8 180.689 13 9.46E-07 14 Residual 9420.8 11 1345.835 Total 495776.5 12 6. Coefficientsulatioh Standard Error 4 t Stat 5 P-value 6 Lower 95% 19 Upper 95% 19 125.289 1 14.1996 2 Intercept 40.9333 3.06 0.0183 28.4969 222.0807 population 0.9100 15.60 1.07E-06 12.0478 16.3517 bus_rating 22.8107 3 5.7692 3.95 0.0055 9.1686 36.4527 (b) The MINITAB output The regression equation is = 125 + 14.2 population + 22.8 bus_rating Jina fashicn incocase Together tiecBur Predictor Coef SE Coef 4 T5 Constant 125.29 1 40.93 3.06 0.018 population 14.1996 2 0.91 15.6 Horg 6 900°0 R-Sq (adj) = 97.6% 9 a io asl bus_rating 22.811 3 OL E 5.769 S = 36.6856 7 R-Sq 98.10% 8 %3D %3D %3D Analysis of Variance Source MS SS Regression 2 486356 10 243178 180.69 13 0.00014 Residual Error 9421 11 1346 Total 495777 12 Predicted Values for New Observations Fit 15 SE Fit 16 95% CI 17 95% PI18 15 (921.0, 992.2) (862.8, 1050.4) 956.6 Values of Predictors for New Observations New Obs population bus rating 47.3 5 1 statistics 6 p-values for / statistics standard error = ST/ 3 b2 9 Adjusted R2 14 p-value for F(model) standard error of the estimate b, 4s 10 Explained variation 1 bo 2 b, || 11 SSE Unexplained variation 12 Total variation 13 F(model) statistic 8 R standard error of the estimate y 15 ý = point prediction when x, L = 'Y pua 18 95% prediction interval when Ir 47.3 and x, = 7 19 95% confidence interval for B, %3D 17 95% confidence interval when x, = 47.3 and x, = 7
Transcribed Image Text:Multiple Regression and Model Building Excel and MINITAB Outputs of a Regression Analysis of the Tasty Sub Shop Revenue Data in Table 14.1 Using the Model y = Bo + B,x, + Bx2 + E FIGURE 14.4 %3D (a) The Excel output Regression Statistics Multiple R 0.9905 R Square 0.9810 8 Adjusted R Square 0.9756 9 36.6856 7 Standard Error Observations (35) ANOVA df SS 486355.7 10 MS F Significance F Regression 243177.8 180.689 13 9.46E-07 14 Residual 9420.8 11 1345.835 Total 495776.5 12 6. Coefficientsulatioh Standard Error 4 t Stat 5 P-value 6 Lower 95% 19 Upper 95% 19 125.289 1 14.1996 2 Intercept 40.9333 3.06 0.0183 28.4969 222.0807 population 0.9100 15.60 1.07E-06 12.0478 16.3517 bus_rating 22.8107 3 5.7692 3.95 0.0055 9.1686 36.4527 (b) The MINITAB output The regression equation is = 125 + 14.2 population + 22.8 bus_rating Jina fashicn incocase Together tiecBur Predictor Coef SE Coef 4 T5 Constant 125.29 1 40.93 3.06 0.018 population 14.1996 2 0.91 15.6 Horg 6 900°0 R-Sq (adj) = 97.6% 9 a io asl bus_rating 22.811 3 OL E 5.769 S = 36.6856 7 R-Sq 98.10% 8 %3D %3D %3D Analysis of Variance Source MS SS Regression 2 486356 10 243178 180.69 13 0.00014 Residual Error 9421 11 1346 Total 495777 12 Predicted Values for New Observations Fit 15 SE Fit 16 95% CI 17 95% PI18 15 (921.0, 992.2) (862.8, 1050.4) 956.6 Values of Predictors for New Observations New Obs population bus rating 47.3 5 1 statistics 6 p-values for / statistics standard error = ST/ 3 b2 9 Adjusted R2 14 p-value for F(model) standard error of the estimate b, 4s 10 Explained variation 1 bo 2 b, || 11 SSE Unexplained variation 12 Total variation 13 F(model) statistic 8 R standard error of the estimate y 15 ý = point prediction when x, L = 'Y pua 18 95% prediction interval when Ir 47.3 and x, = 7 19 95% confidence interval for B, %3D 17 95% confidence interval when x, = 47.3 and x, = 7
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