dit 54%COE Tue Oct 24 11:37PM a e to Blackboard I System Chapter 6, Exercises fo
ID: 3115354 • Letter: D
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dit 54%COE Tue Oct 24 11:37PM a e to Blackboard I System Chapter 6, Exercises for Section 6.3, Question 14cd The data below show the average heights of 12 weeping Higan cherry trees planted in Washington, D.C, as they grew over time At the time of the planting, each of the trees was one year old and 6 feet in height. A logarithmic regression model that represents the data is h -61082 In x + 6.0993 , where x = haverage height of the trees (in feet) age of tree (in Age of Tree (years) Height feet) 6.0 9.5 13.0 15.0 16.5 17.5 18.5 19.0 19.5 19.7 10 19.8 a. Use the model to predict the height of a cherry tree when it is 28 years old. How confident are you in your prediction?Explanation / Answer
The logarithmic regression model representing the data is h = 6.1082ln x + 6.0993 where x is the age of trees (in years)and h is the average height of trees (in feet).
(a). The model predicts that a 28 year old tree will have a height of 6.1082ln28+6.0993 = 6.1082*3.33220451 +6.0993 = 20.35377159+6.0993 = 26.45307159 = 26.5 feet ( on rounding off to 1 decimal place).
(b). As per the given table, the age of a 12 feet high cherry tree is between 2 and 3 years. If we assume the height to be proportional to age, then the age of a 12 feet high tree is average of (2/9.5)*12 and (3/13)*12 i.e. average of 2.5263 and 2.7692 or, 2.65 years . Also, as per the model, when h = 12, we have 12 = 6.1082lnx+6.0993 or, 6.1082lnx = 12-6.0993 or, lnx = 5.9007/6.1082 = 0.966029272 so that x =e0.966029272 = 2.62749067 = 2.63years
Using the table, the estimated age of a 12 feet high cherry tree is 2.6 years ( on rounding off to 1 decimal place).
Using the given regression model, the estimated age of a 12 feet high cherry tree is 2.63 years ( on rounding off to 2 decimal places).
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