12.6 Linear Correlation and Regression 835 41. Sports Drinks The following table shows the energy provided (in kilocalories) and the amount of carbohydrates (in grams) in a 100-m bottle of a sports drink for six different brands. DS Carbohydrates (grams) 6.5 7 6 26.4 6 Energy (kilocalories) 27 30 25 10 28 24 a) Determine the correlation coefficient between amount of carbohydrates and the energy provided. 0.993 b) Determine whether a correlation exists at 0.05. α= Yes c) Determine the equation of the line of best fit for the amount of carbohydrates and the energy provided. y x 3.90 1.94 = + d) Use the equation in part (c) to estimate the amount of energy provided in a sports drink with 5 grams of carbohydrates. 21 kilocalories ≈ 42. Movie Ratings The following table shows the percentage of positive reviews received on a website and the gross earnings, in millions of dollars, for 10 selected movies. $ Percentage of Positive Reviews 95 62 75 75 45 75 87 55 91 60 Earnings (millions of dollars) 100 67 80 120 0 90 60 60 90 100 a) Determine the correlation coefficient between percentage of positive reviews and the movie’s earnings. 0.594 b) Determine whether a correlation exists at 0.05. α= No c) Determine the equation of the line of best fit for the percentage of positive reviews and the movies’ earnings. = + y x 1.20 9.86 d) Use the equation in part (c) to estimate a movie’s earnings if 80% of the movie’s reviews are positive. ≈ $86 million 43. Hitting the Brakes a) Examine the following art. Do you believe that there is a positive correlation, a negative correlation, or no correlation between the speed of a car and the stopping distance when the brakes are applied? Answers will vary. Hitting the brakes At high speed on a wet road, a typical midsize car may need a distance longer than two football fields to come to a stop. Source: CarAndDriver.com Braking distance at given mph 60 mph 140 ft 65 mph 164 ft 70 mph 190 ft 75 mph 218 ft 80 mph 247 ft 60 mph 280 ft 65 mph 410 ft 80 mph 618 ft Wet pavement Dry pavement 75 mph 545 ft 70 mph 475 ft b) Do you believe that there is a stronger correlation between the speed of a car and the stopping distance on wet or dry roads? Answers will vary. c) Use the figure to construct two scatter diagrams, one for dry pavement and the other for wet pavement. Place the speed of the car on the horizontal axis. * d) Compute the correlation coefficient for the speed of the car and the stopping distance for dry pavement. 0.999 e) Repeat part (d) for wet pavement. 0.990 f) Were your answers to parts (a) and (b) correct? Explain. Answers will vary. g) Determine the equation of the line of best fit for dry pavement. y x 5.36 183.40 = − h) Repeat part (g) for wet pavement. y x 16.22 669.80 = − i) Use the equations in parts (g) and (h) to estimate the stopping distance of a car going 77 mph on both dry and wet pavements. Dry, 229.3 ft; wet, 579.1 ft 44. Chlorine in a Swimming Pool A gallon of chlorine is put into a swimming pool. Each hour later for the following 6 hours the percent of chlorine that remains in the pool is measured. The following information is obtained. DS Time (hours) 1 2 3 4 5 6 Chlorine remaining (percent) 80.0 76.2 68.7 50.1 30.2 20.8 *See Instructor Answer Appendix
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