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Unformatted text preview: Midterm 1 Practice Problems 2 With Solutions (1) Regress a Wreck A statistician is trying to learn what factors affect the price of a used car. Her Y variable is the price of the car. She is considering several possible predictor variables. They are X 1 , the original value of the car, X 2 , the mileage on the car, X 3 , the number of repairs that have been done on the car, and X 4 , the number of seat belts in the car. (a) For each of the four possible predictor variables the statistician has obtained the correlation of Y and X, and the covariance of Y and X. Cor ( Y,X 1 ) = . 795 Cov ( Y,X 1 ) = 3 , 688 , 147 Cor ( Y,X 2 ) = . 789 Cov ( Y,X 2 ) = 149 . 155 Cor ( Y,X 3 ) = . 539 Cov ( Y,X 3 ) = 1186 . 4 Cor ( Y,X 4 ) = . 004 Cov ( Y,X 4 ) = 7 . 6 Say what a plot of Y vs X should look like in each case. Solution: Variable X 1 (original value) has a strong positive correlation with Y so the plot should show a clear upward trend. Variable X 4 (number of seatbelts) has a correlation with Y that is close to 0 and so the plot should be nearly flat i.e. not showing a clear relationship between X and Y. Variable X 2 (mileage) has the stronger of the two negative correlations (closer to 1) so the plot should show the stronger of the two downward trends. The points would less spread out about the line than in the plot for X 3 , number of repairs. (b) Rank the variables X 1 ,X 2 ,X 3 ,X 4 in terms of how good a job you expect them to do of pre dicting Y based on the values given in part (a) (NOT on your common sense opinion!) Order them from best predictor to worst predictor and briefly explain your reasoning. Solution: The strength of the relationship is determined by the correlation. (Note: The covariance is not good for comparing strengths of relationships because different units can affect what is a big covariance!) The sign is irrelevant to the strength of the relationshipit only determines the direction of the relationship. Here original price, X 1 , has the highest correlation in absolute value at .795, followed by mileage, X 2 , at .789, repairs, X 3 , at .539, and seatbelts, X 4 , at .004. The stronger the relationship, the better a predictor the variable will be. Therefore original value will be the best predictor followed by mileage, number of repairs, and number of seatbelts. (c) To simplify matters the statistician has fit two regressions, one of price (Y) on mileage ( X 2 ) and one of price (Y) on the number of repairs the car has had ( X 3 ). Printouts for these regressions are given on the following page. Give three numbers from the printouts that tell you which predictor, X 2 or X 3 is doing a better job and briefly explain why that number tells you it is doing a better job....
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 Winter '07
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