APPENDIX D 801 17. a. Exponential: y = 22 31x-12 [or y = 10.629961211.5874012x for an initial value of 1 that doubles every 1.5 years]. b. Exponential: y = 11.49724152211.4194500332x, where 1971 is coded as 1. c. Moore’s law does appear to be working reasonably well. With R2 = 0.991, the model appears to be very good. Chapter 10: Quick Quiz 1. The points appear to approximate a straight-line pattern that rises from left to right. 2. Conclude that there is sufficient evidence to support the claim of a linear correlation between amount of the dinner and the amount of the tip. 3. r = 1 4. y = 0 + 0.20x (or simply y = 0.20x) 5. None of the given values change when the variables are switched. 6. The value of r does not change if all values of one of the variables are multiplied by the same constant. 7. Because r must be between -1 and 1 inclusive, the value of 1.200 is the result of an error in the calculation of r. 8. The best predicted tip is $12.26. It was found by substituting $84.62 for x in the regression equation. 9. The best predicted tip is $9.76. Because there is not sufficient evidence to support the claim of a linear correlation between the cost of dinner and the tip, the best predicted tip is found by computing the mean of the eight sample tips. 10. Because r2 = 0.716, it follows that 0.716 (or 71.6%) of the variation in tips is explained by the linear relationship between amounts of dinner and amounts of tips. It then follows that 0.284 (or 28.4%) of the variation in tips is not explained by the linear relationship between amounts of dinner and amounts of tips. Chapter 10: Review Exercises 1. r = 0.445. P-value: 0.318 (Table: 7 0.05). Critical values: {0.754. There is not sufficient evidence to support the claim that there is a linear correlation between size and revenue. It does not appear that a casino can increase its revenue by enlarging its size. 2. a. y = 63.9 + 0.443x b. Best predicted value of revenue: y = 134.7 million dollars. Because the predicted amount of revenue is 134.7 million dollars for any casino size, the prediction is not likely to be accurate. 3. a. r = 0.450. b. With P@value = 0.192 (Table: 7 0.05) and critical values: r = {0.632 (assuming a 0.05 significance level), there is not sufficient evidence to support the claim that there is a linear correlation between time and height. 13. The best regression equation is HWY = 58.9 - 0.00749 Weight. The three different possible regression equations all have a P-value of 0.000. Given that the single predictor variable of Weight yields an adjusted R2 of 0.787 that is only slightly less than the adjusted R2 of 0.791 obtained by using the two predictor variables of Weight and Displacement, it is better to use the single predictor variable instead of two predictor variables. (The single predictor variable of Displacement has an adjusted R2 of 0.506.) Because the adjusted R2 of 0.787 isn’t very close to 1, it is likely that predicted values will not be very accurate. 15. The best regression equation is yn = 109 - 0.00670x 1, where x1 represents volume. It is best because it has the highest adjusted R2 value of -0.0513 and the lowest P-value of 0.791. The three regression equations all have adjusted values of R2 that are very close to 0, so none of them are good for predicting IQ. It does not appear that people with larger brains have higher IQ scores. 17. For H0: b1 = 0, the test statistic is t = 10.814, the P-value is less than 0.0001, so reject H0 and conclude that the regression coefficient of b1 = 0.769 should be kept. For H0: b2 = 0, the test statistic is t = 29.856, the P-value is less than 0.0001, so reject H0 and conclude that the regression coefficient of b2 = 1.01 should be kept. It appears that the regression equation should include both independent variables of height and waist circumference. 19. yn = 3.06 + 82.4x 1 + 2.91x2, where x1 represents sex and x2 represents age. Female: 61 lb; male: 144 lb. The sex of the bear does appear to have an effect on its weight. The regression equation indicates that the predicted weight of a male bear is about 82 lb more than the predicted weight of a female bear with other characteristics being the same. Section 10-5 1. y = x2. The quadratic model describes the relationship, and R2 = 1. 3. 2.7% of the variation in Super Bowl points can be explained by the exponential model that relates the variable of year and the variable of points scored. Because such a small percentage of the variation is explained by the model, the model is not very useful. 5. The quadratic and power models both yield the same result: d = 0.8t2. 7. Exponential: y = 100011.01x2 9. Quadratic: y = 0.000154x2 + 0.0799x + 6.06, where x is the year with 2000 coded as 1, and y is the world population in billions. 11. Logarithmic: y = 3.22 + 0.293 ln x 13. Quadratic: y = 84.0x2 - 953x + 13,289. (Result is based on the year 2000 coded as 1.) Using the rounded coefficients, the projected value for the last year is 25,506.0, which isn’t too far from the actual value of 26,828.4. Because R2 = 0.925 for the quadratic model, which is high, predicted values are likely to be reasonably accurate, but we should remember that stock market values can be dramatically affected by events that cannot be foreseen by our most creative minds. 15. Power: y = 7.891x-0.3712, where x is the depth and y is the magnitude. The predicted magnitude is 4.82, which is far from the actual magnitude of 7.10. Because R2 = 0.613 for the power model, which isn’t very high, predicted values are not likely to be very accurate.
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