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Problem 5. Formula 1 You wish to establish a linear relationship between the distances driven by two Formula 1 drivers, Driver A and Driver B, during a specific Grand Prix. The distances are given in the table below: Driver A (miles) Driver B (km) 186.4 300 199.5 320 211.3 340 223.7 360 236.1 380 248.5 400 260.9 420 273.4 440 Table 1: Distances driven by Driver A (in miles) and Driver B (in kilometers). Driver A's distances are measured in miles, and Driver B's distances are measured in kilometers. You are asked to model Driver B's distances as a linear function of Driver A's distances. However, you are also interested in converting Driver A's distances into kilometers to make the relationship clearer. a) b) Your friend Zoe suggests that instead of converting all the distances for Driver A from miles to kilometers before performing least squares regression, you could perform the regression with Driver A's distances in miles and then convert the slope and intercept afterward. Recall that distance in miles can be converted to kilometers using the formula: 1 mile == 1.60 km Is Zoe correct that you'll get the same regression coefficients either way? Show your work. More generally, suppose we want to perform least squares regression for a linear relationship y = w₁x+wo, where x is Driver A's distance in miles and y is Driver B's distance in kilometers. How do the slope w₁ and intercept wo of the regression line change if we replace x with a linear transformation f(x) = ax + b, where a and b are constants? Prove your answer by expressing the new slope and intercept in terms of the original slope w₁, intercept wo, and the constants a and b.

Problem 5. Formula 1 You wish to establish a linear relationship between the distances driven by two Formula 1 drivers, Driver A and Driver B, during a specific Grand Prix. The distances are given in the table below: Driver A (miles) Driver B (km) 186.4 300 199.5 320 211.3 340 223.7 360 236.1 380 248.5 400 260.9 420 273.4 440 Table 1: Distances driven by Driver A (in miles) and Driver B (in kilometers). Driver A's distances are measured in miles, and Driver B's distances are measured in kilometers. You are asked to model Driver B's distances as a linear function of Driver A's distances. However, you are also interested in converting Driver A's distances into kilometers to make the relationship clearer. a) b) Your friend Zoe suggests that instead of converting all the distances for Driver A from miles to kilometers before performing least squares regression, you could perform the regression with Driver A's distances in miles and then convert the slope and intercept afterward. Recall that distance in miles can be converted to kilometers using the formula: 1 mile == 1.60 km Is Zoe correct that you'll get the same regression coefficients either way? Show your work. More generally, suppose we want to perform least squares regression for a linear relationship y = w₁x+wo, where x is Driver A's distance in miles and y is Driver B's distance in kilometers. How do the slope w₁ and intercept wo of the regression line change if we replace x with a linear transformation f(x) = ax + b, where a and b are constants? Prove your answer by expressing the new slope and intercept in terms of the original slope w₁, intercept wo, and the constants a and b.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 94E
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Problem 5. Formula 1 You wish to establish a linear relationship between the distances driven by two Formula 1 drivers, Driver A and Driver B, during a specific Grand Prix. The distances are given in the table below: Driver A (miles) Driver B (km) 186.4 300 199.5 320 211.3 340 223.7 360 236.1 380 248.5 400 260.9 420 273.4 440 Table 1: Distances driven by Driver A (in miles) and Driver B (in kilometers). Driver A's distances are measured in miles, and Driver B's distances are measured in kilometers. You are asked to model Driver B's distances as a linear function of Driver A's distances. However, you are also interested in converting Driver A's distances into kilometers to make the relationship clearer. a) b) Your friend Zoe suggests that instead of converting all the distances for Driver A from miles to kilometers before performing least squares regression, you could perform the regression with Driver A's distances in miles and then convert the slope and intercept afterward. Recall that distance in miles can be converted to kilometers using the formula: 1 mile == 1.60 km Is Zoe correct that you'll get the same regression coefficients either way? Show your work. More generally, suppose we want to perform least squares regression for a linear relationship y = w₁x+wo, where x is Driver A's distance in miles and y is Driver B's distance in kilometers. How do the slope w₁ and intercept wo of the regression line change if we replace x with a linear transformation f(x) = ax + b, where a and b are constants? Prove your answer by expressing the new slope and intercept in terms of the original slope w₁, intercept wo, and the constants a and b.
Transcribed Image Text:Problem 5. Formula 1 You wish to establish a linear relationship between the distances driven by two Formula 1 drivers, Driver A and Driver B, during a specific Grand Prix. The distances are given in the table below: Driver A (miles) Driver B (km) 186.4 300 199.5 320 211.3 340 223.7 360 236.1 380 248.5 400 260.9 420 273.4 440 Table 1: Distances driven by Driver A (in miles) and Driver B (in kilometers). Driver A's distances are measured in miles, and Driver B's distances are measured in kilometers. You are asked to model Driver B's distances as a linear function of Driver A's distances. However, you are also interested in converting Driver A's distances into kilometers to make the relationship clearer. a) b) Your friend Zoe suggests that instead of converting all the distances for Driver A from miles to kilometers before performing least squares regression, you could perform the regression with Driver A's distances in miles and then convert the slope and intercept afterward. Recall that distance in miles can be converted to kilometers using the formula: 1 mile == 1.60 km Is Zoe correct that you'll get the same regression coefficients either way? Show your work. More generally, suppose we want to perform least squares regression for a linear relationship y = w₁x+wo, where x is Driver A's distance in miles and y is Driver B's distance in kilometers. How do the slope w₁ and intercept wo of the regression line change if we replace x with a linear transformation f(x) = ax + b, where a and b are constants? Prove your answer by expressing the new slope and intercept in terms of the original slope w₁, intercept wo, and the constants a and b.
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