Question

2. Multiple coefficient of determination Macroeconomics is the study of the economy as a whole. A macroeconomic variable is one that measures a characteristic of the whole economy or one of its large-scale sectors. In forecasting the sales of a product, market researchers frequently use macroeconomic variables in addition to marketing mix variables (marketing mix variables include product, price, place [or distribution], and promotion). A market researcher is analyzing an existing multiple regression model that predicts sales for different brands of digital cameras. The dependent variable is: ymonthly sales of specified digital camera (in thousands of dollars) The independent variables are the following marketing mix variables: X1ratings given by a popular digital photography magazine x2 average sale price (in dollars) X3advertising spending for the given month (in thousands of dollars) The estimated multiple regression equation using data with 33 observations is as follows: y2,364 + 210x1 - 193x2 357x3 The regression just given yields a multiple coefficient of determination of R2-0.45 and an adjusted multiple coefficient of determination of R2. -0.42. The multiple coefficient of determination indicates the proportion of variability in the dependent variable that can be explained by the regression model The researcher would like to improve upon this model by including a macroeconomic variable that may affect sales. He decides to include the following variable: X4 most recent quarterly GDP growth rate The estimated multiple regression equation with the additional independent variable is as follows: - 2,103 + 279x1 145x2 + 313x356x4 The ANOVA table for the new regression model is shown as follows: Analysis of Variance Source of Variation Regression Error Total *Obtained by dividing the respective sums of squares by their corresponding degrees of freedom. For example, the total mean square of 3,637 is the total sum of squares divided by its degrees of freedom, or 116,376/32 Sum of Squares 54,719 61,657 116,376 Degrees of Freedom 4 28 32 Mean Square 13,680 2,202 3,637 F-value P-value 6.21 0.0010 The multiple coefficient of determination, denoted R2, is the ratio of the the estimated multiple regression equation explains 0 . The R2 for the new regres sum of squares due to regression total sum of squares sum of squares due to error ew of the vari The sum of squares due to error divided by the total sum of squares is and 1 minus this ratio is The adjusted multiple coefficient of determination, denoted R2a, for the new regression is The mean square due to error divided by the total mean square is and 1 minus this ratio is In general, adding independent variables to a multiple regression model reduces the The multiple coefficient of determination , and the adjusted multiple coefficient of determination The ANOVA table for the new regression model is shown as follows: Analysis of Variance Source of Variation Regression Error Total *Obtained by dividing the respective sums of squares by their corresponding degrees of freedom. For example, the total mean square of 3,637 is the total sum of squares divided by its degrees of freedom, or 116,376/32 Sum of Squares 54,719 61,657 116,376 Degrees of Freedom 4 28 32 Mean Square* 13,680 2,202 3,637 F-value P-value 6.21 0.0010 The multiple coefficient of determination, denoted R2, is the ratio of the the esti to The R2 for the new regression is ins , indicating that the new total sum of squares sum of squares due to regression sum of squares due to error of the variability of digital camera sales The sum of squares due to error divided by the total sum of squares is , and 1 minus this ratio is The adjusted multiple coefficient of determination, denoted R2a, for the new regression is The mean square due to error divided by the total mean square is and 1 minus this ratio is In general, adding independent variables to a multiple regression model reduces the The multiple coefficient of determination and the adjusted multiple coefficient of determination The ANOVA table for the new regression model is shown as follows: Analysis of Variance Degrees of Freedom 4 28 32 Source of Variation Regression Error Total *Obtained by dividing the respective sums of squares by their corresponding degrees of freedom. For example, the total mean square of 3,637 is the total sum of squares divided by its degrees of freedom, or 116,376/32 Sum of Squares 54,719 61,657 116,376 Mean Square* 13,680 2,202 3,637 F-value P-value 0.0010 6.21 The multiple coefficient of determination, denoted R2, is the ratio of the the estimated multiple regression equation explains | 47% | of the variability of digital camera sales to The R2 for the new regression is , indicating that the new 47% 5590 5390 4590 The sum of squares due to error divided by the , and 1 minus this ratio is squares is The adjusted multiple coefficient of determination, denoted R2a, for the new regression is The mean square due to error divided by the total mean square is and 1 minus this ratio is In general, adding independent variables to a multiple regression model reduces the The multiple coefficient of determination , and the adjusted multiple coefficient of determination The ANOVA table for the new regression model is shown as follows Analysis of Variance Source of Variation Regression Error Total *Obtained by dividing the respective sums of squares by their corresponding degrees of freedom. For example, the total mean square of 3,637 is the total sum of squares divided by its degrees of freedom, or 116,376/32. Sum of Squares 54,719 61,657 116,376 Degrees of Freedom 4 28 32 Mean Square 13,680 2,202 3,637 F-valueP-value 0.0010 6.21 The multiple coefficient of determination, denoted R2, is the ratio of the the estimated multiple regression equation explains 47% of the variability of digital camera sales. to The R2 for the new regression is , indicating that the new The sum of squares due to error divided by the total sum of squares is , and 1 minus this ratio is 0.47 0.43 0.53 0.39,(占sion is The adjusted multiple coefficient of determination, denoted R2a, for th The mean square due to error divided by the total mean square is and 1 minus this ratio is In general, adding independent variables to a multiple regression model reduces the The multiple coefficient of determination , and the adjusted multiple coefficient of determination The ANOVA table for the new regression model is shown as follows Analysis of Variance Source of Variation Regression Error Total Sum of Squares 54,719 61,657 116,376 Degrees of Freedom 4 28 32 Mean Square* 13,680 2,202 3,637 F-value P-value 6.21 0.0010 Obtained by dividing the respective sums of squares by their corresponding degrees of freedom. For example, the total mean square of 3,637 is the total sum of squares divided by its degrees of freedom, or 116,376/32 The multiple coefficient of determination, denoted R2, is the ratio of the The R2 for the new regression is indicating that the new estimated multiple regression equation explains 47% of the variability of digital camera sales. The sum of squares due to error divided by the total sum of squares is , and 1 minus this ratio is 0.57 0.47 0.53 0.61 d multiple coefficient of determination, denoted R2a, for the new regression is The mean square due to error divided by the total mean square is and 1 minus this ratio is In general, adding independent variables to a multiple regression model reduces the The multiple coefficient of determination and the adjusted multiple coefficient of determination The ANOVA table for the new regression model is shown as follows: Analysis of Variance Degrees of Freedom 4 28 32 Source of Variation Regression Error Total *Obtained by dividing the respective sums of squares by their corresponding degrees of freedom. For example, the total mean square of 3,637 is the total sum of squares divided by its degrees of freedom, or 116,376/32 Sum of Squares 54,719 61,657 116,376 Mean Square* 13,680 2,202 3,637 F-value P-value 6.21 0.0010 The multiple coefficient of determination, denoted R2, is the ratio of the the estimated multiple regression equation explains 47% of the variability of digital camera sales. to . The R2 for the new regression is , indicating that the new The sum of squares due to error divided by the total sum of squares is and 1 minus this ratio is The adjusted multiple coefficient of determination, denoted R2a, for the new regression is 0.61 0.42 0.58 0.39 The mean square due to error divided by the total mean square is and 1 mi IS In general, adding independent variables to a multiple regression model reduces the The multiple coefficient of determination , and the adjusted multiple coefficient of determination The ANOVA table for the new regression model is shown as follows: Analysis of Variance Degrees of Freedom 4 28 32 Source of Variation Regression Error Sum of Squares 54,719 61,657 116,376 Mean Square* 13,680 2,202 3,637 F-value P-value 6.21 0.0010 *Obtained by dividing the respective sums of squares by their corresponding degrees of freedom. For example, the total mean square of 3,637 is the total sum of squares divided by its degrees of freedom, or 116,376/32 The multiple coefficient of determination, denoted R2, is the ratio of the the estimated multiple regression equation explains 47% of the variability of digital camera sales to . The R2 for the new regression is , indicating that the new The sum of squares due to error divided by the total sum of squares is and 1 minus this ratio is The adjusted multiple coefficient of determination, denoted R2a, for the new regression is The mean square due to error divided by the total mean square is , and 1 minus this ratio is 0.61 0.47 0.39 0.43ation In general, adding independent variables to a multiple regression es the The multiple coefficient and the adjusted multiple coefficient of determination The ANOVA table for the new regression model is shown as follows Analysis of Variance Source of Variation Regression Error Total *Obtained by dividing the respective sums of squares by their corresponding degrees of freedom. For example, the total mean square of 3,637 is the total sum of squares divided by its degrees of freedom, or 116,376/32 Sum of Squares 54,719 61,657 116,376 Degrees of Freedom 4 28 32 Mean Square 13,680 2,202 3,637 F-value P-value 6.21 0.0010 The multiple coefficient of determination, denoted R2, is the ratio of the the estimated multiple regression equation explains 47% of the variability of digital camera sales. to The R2 for the new regression is , indicating that the new The sum of squares due to error divided by the total sum of squares is and 1 minus this ratio is The adjusted multiple coefficient of determination, denoted R-a, for the new regression is The mean square due to error divided by the total mean square is and 1 minus this ratio is In general, adding independent variables to a multiple regression model reduces the . The multiple coefficient of determination and the adjusted multiple coefficient of determination total sum of squares sum of squares due to regression sum of squares due to error The ANOVA table for the new regression model is shown as follows: Analysis of Variance Source of Variation Regression Error Total Obtained by dividing the respective sums of squares by their corresponding degrees of freedom. For example, the total mean square of 3,637 is the total sum of squares divided by its degrees of freedom, or 116,376/32 Sum of Squares 54,719 61,657 116,376 Degrees of Freedom 4 28 32 Mean Square* 13,680 2,202 3,637 F-value P-value 6.21 0.0010 The multiple coefficient of determination, denoted R2, is the ratio of the to The R2 for the new regression is , indicating that the new estimated multiple regression equation explains 47% of the variability of digital camera sales. The sum of squares due to error divided by the total sum of squares is and 1 minus this ratio is The adjusted multiple coefficient of determination, denoted R2a, for the new regression is The mean square due to error divided by the total mean square is , and 1 minus this ratio is In general, adding independent variables to a multiple regression model reduces the The multiple coefficient of determination , and the adjusted multiple coefficient of determination increases could either increase or decrease decreases The ANOVA table for the new regression model is shown as follows: Analysis of Variance Source of Variation Regression Error Total *Obtained by dividing the respective sums of squares by their corresponding degrees of freedom. For example, the total mean square of 3,637 is the total sum of squares divided by its degrees of freedom, or 116,376/32 Sum of Squares 54,719 61,657 116,376 Degrees of Freedom 4 28 32 Mean Square* 13,680 2,202 3,637 F-value P-value 6.21 0.0010 The multiple coefficient of determination, denoted R2, is the ratio of the estimated multiple regression equation explains 47% of the variability of digital camera sales. the . The R2 for the new regression is , indicating that the new The sum of squares due to error divided by the total sum of squares is , and 1 minus this ratio is The adjusted multiple coefficient of determination, denoted R2a, for the new regression is The mean square due to error divided by the total mean square is , and 1 minus this ratio is In general, adding independent variables to a multiple regression model reduces the The multiple coefficient of determination and the adjusted multiple coefficient of determination decreases increases could either increase or decrease

Answer 1

The multiple coef ficient of determination, denoted R2, is the ratio of the sum of squares due to regression to the total sum of squares. The R2 for the new regression is (54719/116376)*100 = 47% of the variability of digital camera sales. The sum of square due to error divided by the total sum of squares is (61657/116376) 0.53, and 1 minus this ratio is 0.47. The adjusted multiple coef ficient of determination, denoted R2 for new regression is 1 - (32/28) 0.53 0.39. The mean square due to error divided by the total mean square is (2202/3637) 0.61, and 1 minus this ratio is 0.39. In general, adding independent variables to a multiple regression model reduces the sum of squares due to error. The multiple coe f ficient of determination increases and the adjusted multiple coef ficient of determination could either increase or decrease