Ch. 12 Exercises Answers (Simple Regression Analysis and Correlation) PDF

Ch. 12. Simple Regression Analysis and Correlation 1) Use the following data. x 53 47 41 50 58 62 45 60 y 5 5 7 4 10 12 3 11 a. Determine the equation of the simple regression...

Ch. 12. Simple Regression Analysis and Correlation 1) Use the following data. x 53 47 41 50 58 62 45 60 y 5 5 7 4 10 12 3 11 a. Determine the equation of the simple regression line to predict y from x. b. Using the x values, solve for the predicted values of y and the residuals. c. What is the standard error of the estimated slope? d. Test the slope of the regression line. What do you conclude about the slope? e. What is the coefficient of determination? Answer SUMMARY OUTPUT Regression Statistics Multiple R 0.780 R Square 0.608 Adjusted R Square 0.543 Standard Error 2.326 Observations 8 ANOVA df SS MS F Significance F Regression 1 50.41 50.41 9.316494687 0.022445769 Residual 6 32.465 5.410833333 Total 7 82.875 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept -11.335 6.104 -1.857 0.113 -26.270 3.600 x 0.355 0.116 3.052 0.022 0.070 0.640 a) ŷ = -11.335 + 0.355 x b) ŷ (Predicted Values) (y- ŷ ) residuals 7.48 -2.48 5.35 -0.35 3.22 3.78 6.415 -2.415 9.255 0.745 10.675 1.325 4.64 -1.64 9.965 1.035 c) SE(beta_hat) = 0.116 d) Ho:  = 0  =.05 Ha:   0 p = 0.022 < 0.05 =  => Reject Ho. The population slope is different from zero. There is a relation between x and y. e) r2 = 0.608. 60.8 % of the variation in Y can be explained by the variation in x. 1  2) Determine the equation of the trend line through the following cost data. Use the equation of the line to forecast cost for year 7. Year Cost ($ millions) 1 56 2 54 3 49 4 46 5 45 Answer SUMMARY OUTPUT Regression Statistics Multiple R 0.978 R Square 0.957 Adjusted R Square 0.943 Standard Error 1.155 Observations 5 ANOVA df SS MS F Significance F Regression 1 90 90 67.5 0.003774196 Residual 3 4 1.333333333 Total 4 94 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 59 1.211 48.718 0.000 55.146 62.854 Year -3 0.365 -8.216 0.004 -4.162 -1.838 ̂ = 59 – 3Year 𝐶𝑜𝑠𝑡 ̂ = 59 – 3Year = 59 – 37 = 59 – 21 = 38 Cost in year 7: 𝐶𝑜𝑠𝑡 2  3) It seems logical that restaurant chains with more units (restaurants) would have greater sales. This assumption is mitigated, however, by several possibilities: some units may be more profitable than others, some units may be larger, some units may serve more meals, some units may serve more expensive meals, and so on. The data shown here were published by Technomic. Perform a simple regression analysis to predict a restaurant chain’s sales by its number of units. How strong is the relationship? Restaurant Y X Name ($ billions) (1000) Chain Sales Number of Units McDonald’s 17.1 12.4 Burger King 7.9 7.5 Taco Bell 4.8 6.8 Pizza Hut 4.7 8.7 Wendy’s 4.6 4.6 KFC 4 5.1 Subway 2.9 11.2 Dairy Queen 2.7 5.1 Hardee’s 2.7 2.9 Answer Regression Statistics Multiple R 0.636 R Square 0.405 Adjusted R Square 0.320 Standard Error 3.764 Observations 9 ANOVA df SS MS F Significance F Regression 1 67.395 67.39503432 4.758 0.066 Residual 7 99.154 14.16483637 Total 8 166.549 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept -0.864 3.265 -0.265 0.799 -8.584 6.856 Number of Units 0.920 0.422 2.181 0.066 -0.077 1.918 Ho:  = 0  =.05 Ha:   0 p = 0.066 < 0.05 =  => Cannot reject Ho. The population slope is not different from zero. There is no relation between sales and number of units. 3  4) Study the following Minitab output from a regression analysis to predict y from x. a. What is the equation of the regression model? b. What is the meaning of the coefficient of x? c. What is the result of the test of the slope of the regression model? Let α=0.05. Why is the t ratio negative? d. Comment on r2 and the standard error of the estimate. e. Comment on the relationship of the F value to the t ratio for x. f. The correlation coefficient for these two variables is -.7918. Is this result surprising to you? Why or why not? Regression Analysis: Y versus X The regression equation is Y = 67.2 – 0.0565 X Predictor Coef SE Coef T p Constant 67.231 5.046 13.32 0.000 X –0.05650 0.01027 –5.50 0.000 S = 10.32 R-Sq = 62.7% R-Sq(adj) = 60.6% Analysis of Variance Source DF SS MS F P Regression 1 3222.9 3222.9 30.25 0.000 Residual Error 18 1918.0 106.6 Total 19 5141.0 Answer a) Y_hat = 67.2 – 0.0565X b) Meaning of beta_hat. When x increases by 1 unit, Y is reduced by 0.0565 c) Ho:  = 0  =.05 Ha:   0 p < 0.0001 < 0.05 =  => Reject Ho. The population slope is different from zero. There is a relation between x and y. d) r2 is.627 or 62.7% of the variability of y is accounted for by x. This is only a modest proportion of predictability. The standard error of the estimate is 10.32. This is best interpreted in light of the data and the magnitude of the data. e) Since only one variable => t2 = F. f) The square root of r2 which is.627 yields.7906 which is the magnitude of the value of r considering rounding error. The negative is not a surprise because the slope of the regression line is also negative indicating an inverse relationship between x and y 4