Adjusted R 2 R 2 never decreases when a new x variable is added to

# Adjusted r 2 r 2 never decreases when a new x

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What is the net effect of adding a new variable? x Did the new x variable add enough explanatory power to offset the loss of one degree of freedom?
Chap 15-26 Shows the proportion of variation in y explained by all x variables adjusted for the number of x variables used (where n = sample size, k = number of independent variables) Penalize excessive use of unimportant independent variables Smaller than R 2 Useful in comparing among models Adjusted R 2 (continued) - - - - - = 1 k n 1 n ) R 1 ( 1 R 2 2 A
Chap 15-27 Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error 47.46341 Observations 15 ANOVA df SS MS F Significance F Regression 2 29460.027 14730.013 6.53861 0.01201 Residual 12 27033.306 2252.776 Total 14 56493.333 .44172 R 2 A = 44.2% of the variation in pie sales is explained by the variation in price and advertising, taking into account the sample size and number of independent variables Multiple Coefficient of Determination (continued)
Chap 15-28 Is the Model Significant? F-Test for Overall Significance of the Model Shows if there is a linear relationship between all of the x variables considered together and y Use F test statistic Hypotheses: H 0 : β 1 = β 2 = … = β k = 0 (no linear relationship) H A : at least one β i ≠ 0 (at least one independent variable affects y)
Chap 15-29 F-Test for Overall Significance Test statistic: where F has (numerator) D 1 = k and (denominator) D 2 = (n – k – 1) degrees of freedom (continued) MSE MSR 1 k n SSE k SSR F = - - =
Chap 15-30 6.5386 2252.8 14730.0 MSE MSR F = = = Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error 47.46341 Observations 15 ANOVA df SS MS F Significance F Regression 2 29460.027 14730.013 6.53861 0.01201 Residual 12 27033.306 2252.776 Total 14 56493.333 (continued) F-Test for Overall Significance With 2 and 12 degrees of freedom P-value for the F-Test
Chap 15-31 H 0 : β 1 = β 2 = 0 H A : β 1 and β 2 not both zero α = .05 df 1 = 2 df 2 = 12 Test Statistic: Decision: Conclusion: Reject H 0 at α = 0.05 The regression model does explain a significant portion of the variation in pie sales (There is evidence that at least one independent variable affects y ) 0 α = .05 F .05 = 3.885 Reject H 0 Do not reject H 0 6.5386 MSE MSR F = = Critical Value: F α = 3.885 F-Test for Overall Significance (continued) F
Chap 15-32 Are Individual Variables Significant? Use t-tests of individual variable slopes Shows if there is a linear relationship between the variable x i and y Hypotheses: H 0 : β i = 0 (no linear relationship) H A : β i ≠ 0 (linear relationship does exist between x i and y)
Chap 15-33 Are Individual Variables Significant? H 0 : β i = 0 (no linear relationship) H A : β i ≠ 0 (linear relationship does exist between x i and y ) Test Statistic: ( df = n – k – 1) i b i s 0 b t - = (continued)
Chap 15-34 Regression Statistics Multiple R 0.72213 R Square 0.52148 Adjusted R Square 0.44172 Standard Error 47.46341 Observations 15 ANOVA df SS MS F Significance F Regression 2 29460.027 14730.013 6.53861 0.01201 Residual 12 27033.306 2252.776 Total 14 56493.333 t-value for Price is t = -2.306, with p-value .0398 t-value for Advertising is t = 2.855, with p-value .0145 (continued) Are Individual Variables Significant?
Chap 15-35 d.f. = 15-2-1 = 12 α = .05 t α /2 = 2.1788 Inferences about the Slope: t Test Example H 0 : β i = 0 H A : β i 0 The test statistic for each variable falls in the rejection region (p-values < .05)

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