Unit 4 Section 4 Exercises MY ANSWERS (1)

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Elementary Statistics Unit 4 : Section 4 Exercises Your solutions should be clear, complete, and sufficiently detailed in order to demonstrate your understanding and communicate your reasoning and method of solving the problem. Exercise 1 The height (in feet) and trunk circumference (in inches) at breast height (4.5 feet above the ground) was measured for a random sample of Eucalyptus trees. The data are summarized below. Trunk Circumference 21.1 20.8 22.5 19.4 23.6 19.8 21.6 19.9 Tree Height 34.2 32.7 35.0 31.9 36.5 31.2 33.8 31.4 (a) Determine the linear regression model that will best predict the height of a Eucalyptus tree based on its trunk circumference at breast height. (b) How well does the linear regression model fit this sample data? (c) Predict the height of a Eucalyptus tree with a trunk circumference at breast height of 22 inches. Page 1 Mathematics 13 L1 (xi) = TRUNK CIRCUMFERENCE L2 (yi) = TREE HEIGHT LinReg(ax+b) L1,L2 ON CALCULATOR GIVES US ANSWER: The linear regression model is yˆ = 1.250x + 6.989 r2= 0.928 Yes, the linear regression model fits the sample well because the coefficitent of determination r2 = 0.928 is quite close to 1. For x = 22, yˆ = 1.250 x (22) + 6.989 = 34.49 This means, the predicted height of a eucalyptus tree with a trunk circumference at breast height of 22 inches would be about 34.5 feet tall.

Elementary Statistics Unit 4 : Section 4 Exercises Exercise 2 A medical researcher wishes to determine how the dosage (in milliliters) of an experimental drug affects the heart rate (in beats per minute) of patients with an elevated heart rate. The data for a sample of eight patients with an elevated heart rate are provided in the following table. Drug Dosage 0 5 10 20 25 30 40 50 Heart Rate 135 124 106 89 85 72 68 62 (a) Determine the linear regression model that will best predict a patient’s heart rate based on the dosage of the drug received. (b) How well does the linear regression model fit this sample data? (c) If a patient with an elevated heart rate is administered a 35 ml dose of this drug, predict the resulting heart rate of the patient. Page 2 Mathematics 13 L1 (xi) = DRUG DOSAGE L2 (yi) = HEART RATE LinReg(ax+b) L1,L2 ON CALCULATOR GIVES US ANSWER: The linear regression model is yˆ = -1.48x + 125.92 r2= 0.920 Yes, the linear regression model fits the sample well because the coefficient of determination r2 = 0.920 is quite close to 1. For x = 35, yˆ = -1.48 x (35) + 125.92 = 74.1 This means, if a patient with an elevated heart rate is administered a 35 ml dose of this drug, the predicted resulting heart rate of this particular patient would be predicted to be around 74 BPM (beats per minute).

Elementary Statistics Unit 4 : Section 4 Exercises Exercise 3 A random sample of 8 people had their height (in inches) and their weight (in pounds) recorded. The resulting data are presented in the table below. Height 71.25 64.5 74 65.75 68.25 63 70.5 66.25 Weight 192 106 227 148 179 132 153 121 (a) Determine the linear regression model that will best predict a person’s weight based on their height. (b) How well does the linear regression model fit this sample data? (c) Predict the weight of a person that is 67 inches tall. Page 3 Mathematics 13 L1 (xi) = HEIGHT L2 (yi) = WEIGHT LinReg(ax+b) L1,L2 ON CALCULATOR GIVES US ANSWER: The linear regression model is yˆ = 9.3274x - 476.4298 r2=0.761 Yes, the linear regression model fits the sample well because the coefficient of determination r2 = 0.761 is pretty close to 1. For x = 67, yˆ = 9.3274 x (67) - 476.4298 = 148.506 So, the predicted weight of a person who is 67 inches tall would be around 148.5 pounds.

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Elementary Statistics Unit 4 : Section 4 Exercises Exercise 4 The following table displays the EPA fuel efficiency estimates (in miles per gallon) and the curb weight (in pounds) for a random sample of current year model vehicles. MPG 23 18 28 19 25 17 18 14 Weight 3184 3598 2734 4082 2623 4685 4178 5488 (a) Determine the linear regression model that will best predict the EPA fuel efficiency estimates (MPG) of a vehicle based on its curb weight. (b) How well does the linear regression model fit this sample data? (c) For a vehicle that weighs 4000 pounds, predict its EPA fuel efficiency estimate. Page 4 Mathematics 13 L1 (xi) = HEIGHT L2 (yi) = WEIGHT LinReg(ax+b) L1,L2 ON CALCULATOR GIVES US ANSWER: The linear regression model is yˆ = -0.004375x + 36.97 r2=0.858 Yes, the linear regression model fits the sample well because the coefficient of determination r2 = 0.858 is pretty close to 1. For x = 4000, yˆ = -0.004375 x (4000) + 36.97 = 19.47 In conclusion, a vehicle that weighs 400 pounds has a predicted EPA fuel efficiency rate of 19 or 20 MPG.