800 APPENDIX D Section 10-3 1. The value of se = 16.27555 cm is the standard error of estimate, which is a measure of the differences between the observed weights and the weights predicted from the regression equation. It is a measure of the variation of the sample points about the regression line. 3. The coefficient of determination is r2 = 0.155. We know that 15.5% of the variation in weight is explained by the linear correlation between height and weight, and 84.5% of the variation in weight is explained by other factors and>or random variation. 5. r2 = 0.089. 8.9% of the variation in tips is explained by the linear correlation between times and tips, and 91.1% of the variation in tips is explained by other factors and>or random variation. 7. r2 = 0.972. 97.2% of the variation in fares is explained by the linear correlation between distances and fares, and 2.8% of the variation in fares is explained by other factors and>or random variation. 9. r = -0.788. Critical values: r = {0.576, assuming a 0.05 significance level. There is sufficient evidence to support a claim of a linear correlation between weights of large cars and the highway fuel consumption amounts. 11. 29.0 mi>gal 13. 27.9 mi>gal 6 y 6 37.7 mi>gal 15. 24.2 mi>gal 6 y 6 36.9 mi>gal 17. a. 10,626.59 b. 68.83577 c. 38.0°F 6 y 6 60.4°F 19. a. 352.7278 b. 109.3722 c. 71.09°F 6 y 6 88.71°F 21. 76.1 million tickets 6 y 6 120 million tickets Section 10-4 1. The response variable is Speed (the mean speed of the winner) and the predictor variables are Distance, the number of Stages, and the number of Finishers. 3. The unadjusted R2 increases (or remains the same) as more variables are included, but the adjusted R2 is adjusted for the number of variables and sample size. The unadjusted R2 incorrectly suggests that the best multiple regression equation is obtained by including all of the available variables, but by taking into account the sample size and number of predictor variables, the adjusted R2 is much more helpful in weeding out variables that should not be included. 5. Son = 18.0 + 0.504 Father+0.277 Mother 7. P-value less than 0.0001 is low, but the values of R210.36492 and adjusted R210.35522 are not high. Although the multiple regression equation fits the sample data best, it is not a good fit, so it should not be used for predicting the height of a son based on the height of his father and the height of his mother. 9. The weight of discarded paper, because it has the best combination of small P-value (0.000) and highest adjusted R2 (0.411). 11. PLAS = -0.170 + 0.290 METAL + 0.122 PAPER + 0.0777 GLASS. That equation has a low P-value of 0.000 and its adjusted R2 value of 0.540 is the largest and it is substantially higher than any of the other values of adjusted R2. 37. With n = 703, there are 701 degrees of freedom. From Table A-3 use the closest t value of 1.965 in the given formula to get the critical values of {0.074. Using a more accurate value of t = 1.963354 from technology leads to the same critical values of {0.074. Section 10-2 1. a. yn represents the predicted value of highway fuel consumption. b. Slope: -0.00749; y-intercept: 58.9 c. The predictor variable is weight which is represented by x. d. 36.4 mi>gal 3. a. A residual is a value of y - yn, which is the difference between an observed value of y and a predicted value of y. b. The regression line has the property that the sum of squares of the residuals is the lowest possible sum. 5. With no significant linear correlation, the best predicted value is y = 37.3 mi>gal. 7. With a significant linear correlation, the best predicted value is 92.0 kg. 9. yn = 3.00 + 0.500x. The data have a pattern that is not a straight line. 11. a. yn = 0.264 + 0.906x b. yn = 2 + 0x 1or yn = 22 c. The results are very different, indicating that one point can dramatically affect the regression equation. 13. yn = 7.97 + 0.0756x. Best predicted value: y = 25.6 million tickets. The best predicted value is very different from the actual value of 90 million tickets that were sold. 15. yn = 1.06 + 0.0452x. Best predicted value: y = $1.68. The best predicted value is very different from the actual tip of $4.55. 17. yn = 5.19 + 2.70x. Best predicted value: $13.55 1or $13.562. The best predicted value is close to the actual fare of $15.30. 19. yn = 50.0 - 0.0886x. Best predicted value: y = 46.4 years. The best predicted value isn’t close to the actual value of 60 years. 21. yn = 0.0329 + 0.969x. Best predicted value: $3.91. 23. yn = 350 + 5.21x. Best predicted value: 1772 mm. The best predicted height is close to the actual height. 25. yn = 0.923 + 0.00665x. Best predicted value: y = 57 points. The best predicted value isn’t close to the actual value of 37 points. 27. yn = 16.5 - 0.00282x. Best predicted value: 15.1 fatalities per 100,000 population. Common sense suggests that the prediction doesn’t make much sense. 29. yn = 0.174 + 0.116x. Best predicted value: $2.49. (Unlike Exercise 15, this larger data set results in a significant linear correlation, so the predicted value is not y.) The best predicted value isn’t very close to the actual tip of $4.55. 31. yn = 5.95 + 2.86x. Best predicted value: $14.80 (or $14.82). The best predicted value is close to the actual fare of $15.30. 33. a. 6.784, 4.802, -0.300, -1.598, -1.248, -2.420, 0.364, 1.670, -7.470 b. 137.862 c. Using yn = -10.0 + 0.200x, the sum of squares of the residuals is 535.560, which is larger than 137.862, which is the sum of squares of the residuals for the regression line.
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