MAT 120 Section 9.3 Measures of Regression

Find the equation of the regression line for the given data. Then construct a scatter plot of the data and draw the regression line. The table shows the shoe size and heights​ (in.) for 6 men. Shoe size, x | Height, y 7.5 | 65.0 9.0 | 66.0 11.0 | 71.0 11.5 | 74.0 13.0 | 74.0 13.5 | 72.0 Find the regression equation. ŷ = ___x + (___).

ANSWER: ŷ = 1.523x + (53.711).

The number of initial public offerings of stock issued in a​ 10-year period and the total proceeds of these offerings​ (in millions) are shown in the table. The equation of the regression line is ŷ = 47.569x + 18,702.99. Complete all parts. Issues, x | Proceeds, y 425 | 19,326 450 | 29,384 690 | 42,680 484 | 31,820 476 | 66,984 400 | 66,521 58 | 21,064 62 | 10,894 187 | 31,603 152 | 27,726 (a) Find the coefficient of determination. Round to 3 decimal places. (b) How can the coefficient of determination be​ interpreted? A. The coefficient of determination is the fraction of the variation in proceeds that is unexplained and is due to other factors or sampling error. The remaining fraction of the variation is explained by the variation in issues. B. The coefficient of determination is the fraction of the variation in proceeds that can be explained by the variation in issues. The remaining fraction of the variation is unexplained and is due to other factors or to sampling error.

ANSWER: (a) r^2 = 0.283. (b) B. The coefficient of determination is the fraction of the variation in proceeds that can be explained by the variation in issues. The remaining fraction of the variation is unexplained and is due to other factors or to sampling error.

The coefficient of determination r(squared) is the ratio of which two types of​ variations? What does r(squared) measure? What does 1−r2 measure? (a) The coefficient of determination is the ratio of the explained variation to the total variation. A. The coefficient of determination is the ratio of the explained variation to the total variation. B. The coefficient of determination is the ratio of the unexplained variation to the total variation. C. The coefficient of determination is the ratio of the total variation to the unexplained variation. D. The coefficient of determination is the ratio of the total variation to the explained variation. (b) Choose the correct answer below. A. The coefficient of determination is the percent of the predicted values that equal the actual data values. B. The coefficient of determination is the percent of variation of y that is explained by the relationship between x and y. C. The coefficient of determination is a measure of how closely two variables vary together. D. The coefficient of determination is the percent of the variation that is unexplained. (c) Choose the correct answer below. A. The value 1 − r^2 is a measure of how closely two variables vary together. B. The value 1 − r^2 is the percent of the predicted values that equal the actual data values. C. The value 1 − r^2 is the percent of variation of y that is explained by the relationship between x and y. D. The value 1 − r^2 is the percent of the variation that is unexplained.

ANSWER: (a) A. The coefficient of determination is the ratio of the explained variation to the total variation. (b) B. The coefficient of determination is the percent of variation of y that is explained by the relationship between x and y. The coefficient of​ determination, r​^2, is the percent of variation of y that is explained by the relationship between x and y. (c) D. The value 1 − r^2 is the percent of the variation that is unexplained. The coefficient of​ determination, r^2​, is the percent of variation of y that is explained by the relationship between x and y.​ Therefore, 1 − r^2​, is the percent of variation of y that is unexplained by the relationship between x and y.

Use the value of the linear correlation coefficient to calculate the coefficient of determination. What does this tell you about the explained variation of the data about the regression​ line? About the unexplained​ variation? r = 0.062 (a) Calculate the coefficient of determination. (r^2 = ?) (b) What does this tell you about the explained variation of the data about the regression​ line? ____% of the variation can be explained by the regression line. About the unexplained​ variation? ____% of the variation is unexplained and is due to other factors or to sampling error.

ANSWER: (a) r^2 = 0.004 (b) 0.4% (c) 99.6% 1 - 0.004 = 0.996

Use the value of the linear correlation coefficient to calculate the coefficient of determination. What does this tell you about the explained variation of the data about the regression​ line? About the unexplained​ variation? r = 0.208 (a) Calculate the coefficient of determination. (r^2 = ?) (b) What does this tell you about the explained variation of the data about the regression​ line? ____% of the variation can be explained by the regression line. (c) About the unexplained​ variation? ___ % of the variation is unexplained and is due to other factors or to sampling error.

ANSWER: (a) r^2 = 0.043 (b) 4.3% (c) 95.7% 1 - 0.043 = 0.957

The table shows the total square footage​ (in billions) of retailing space at shopping centers and their sales​ (in billions of​ dollars) for 10 years. The equation of the regression line is ŷ = 527.983x − 1751.257. Complete all parts. Total Square Footage, x | Sales, y 4.9 | 872.3 5.1 | 929.6 5.2 | 998.5 5.3 | 1065.5 5.6 | 1110.2 5.6 | 1207.9 5.9 | 1296.4 5.9 | 1347.4 5.9 | 1427.5 6.1 | 1535.2 (a) Find the coefficient of determination. round to 3 decimal places. (r^2 = ?) (b) How can the coefficient of determination be​ interpreted? A. The coefficient of determination is the fraction of the variation in sales that can be explained by the variation in total square footage. The remaining fraction of the variation is unexplained and is due to other factors or to sampling error. B. The coefficient of determination is the fraction of the variation in sales that is unexplained and is due to other factors or sampling error. The remaining fraction of the variation is explained by the variation in total square footage.

ANSWER: (a) r^2 = 0.945 (b) A. The coefficient of determination is the fraction of the variation in sales that can be explained by the variation in total square footage. The remaining fraction of the variation is unexplained and is due to other factors or to sampling error.

The money raised and spent​ (both in millions of​ dollars) by all congressional campaigns for 8 recent​ 2-year periods are shown in the table. The equation of the regression line is ŷ = 0.974x − 2.033. Complete all parts. Money raised,x | Money spent,y 472.8 | 436.6 658.4 | 683.3 731.1 | 731.9 787.8 | 751.1 784.4 | 729.2 1027.8 | 1011.3 965.3 | 935.9 1211.7 | 1172.3 (a) Find the coefficient of determination. (r^2 = ?) (b) How can the coefficient of determination be​ interpreted? A. The coefficient of determination is the fraction of the variation in money spent that can be explained by the variation in money raised. The remaining fraction of the variation is unexplained and is due to other factors or to sampling error. B. The coefficient of determination is the fraction of the variation in money spent that is unexplained and is due to other factors or sampling error. The remaining fraction of the variation is explained by the variation in money raised.

ANSWER: (a) r^2 = 0.988 (b) A. The coefficient of determination is the fraction of the variation in money spent that can be explained by the variation in money raised. The remaining fraction of the variation is unexplained and is due to other factors or to sampling error.

The money raised and spent​ (both in millions of​ dollars) by all congressional campaigns for 8 recent​ 2-year periods are shown in the table. The equation of the regression line is ŷ = 0.947x + 26.167. Complete all parts. x | y Money raised | Money spent 461.3 | 448.2 659.1 | 676.1 729.7 | 728.6 781.7 | 774.1 783.1 | 742.4 1040.1 | 1001.1 961.8 | 939.4 1206.5 | 1171.5 (a) Find the coefficient of determination and interpret the result. (r^2 = ?) How can the coefficient of determination be​ interpreted? A. The coefficient of determination is the fraction of the variation in money spent that is unexplained and is due to other factors or sampling error. The remaining fraction of the variation is explained by the variation in money raised. B. The coefficient of determination is the fraction of the variation in money spent that can be explained by the variation in money raised. The remaining fraction of the variation is unexplained and is due to other factors or to sampling error.

ANSWER: (a) r^2 = 0.995 (b) B. The coefficient of determination is the fraction of the variation in money spent that can be explained by the variation in money raised. The remaining fraction of the variation is unexplained and is due to other factors or to sampling error.

Given a set of data and a corresponding regression​ line, describe all values of x that provide meaningful predictions for y. Choose the correct answer below. A. Prediction values are meaningful only for​ x-values that are not included in the original data set. B. Prediction values are meaningful only for​ x-values in​ (or close​ to) the range of the original data. C. Prediction values are meaningful for all​ x-values that are realistic in the context of the original data set.

ANSWER: B. Prediction values are meaningful only for​ x-values in​ (or close​ to) the range of the original data. The equation of a regression line can be used to predict​ y-values over the range of the data. That​ is, prediction values are meaningful only for​ x-values in​ (or close​ to) the range of the data.