A statistics professor bought a new car for $35,000. For the next 5 years, she used several automotive web pages to estimate the value of the car. She then found the least-squares regression line for the data to be ŷ = 35035.71 - 4142.86x where y is the value of the car and is the years since she purchased it. Predict the value of her vehicle after 6 years.
A statistics professor bought a new car for $35,000. For the next 5 years, she used several automotive web pages to estimate the value of the car. She then found the least-squares regression line for the data to be ŷ = 35035.71 - 4142.86x where y is the value of the car and is the years since she purchased it. Predict the value of her vehicle after 6 years.
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![**Title: Estimating Car Depreciation Using Least-Squares Regression**
**Introduction:**
A statistics professor bought a new car for $35,000. For the next 5 years, she used several automotive web pages to estimate the value of the car. She then found the least-squares regression line for the data to be:
\[ \hat{y} = 35035.71 - 4142.86x \]
where \( y \) is the value of the car and \( x \) is the years since she purchased it. Predict the value of her vehicle after 6 years.
**Discussion:**
The equation provided is a linear regression model that describes the depreciation of the car's value over time. The term \( 35035.71 \) represents the initial value of the car when \( x = 0 \), which is close to the purchase price of $35,000. The slope \( -4142.86 \) indicates the annual decrease in the car's value.
**Calculation:**
To predict the value of the car after 6 years, substitute \( x = 6 \) into the regression equation:
\[ \hat{y} = 35035.71 - 4142.86 \times 6 \]
\[ \hat{y} = 35035.71 - 24857.16 \]
\[ \hat{y} = 10178.55 \]
Therefore, the predicted value of the car after 6 years is approximately $10,178.55.
This regression model can be useful for understanding how quickly assets like cars depreciate over time, allowing for better financial planning and decision-making.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F23f14c8c-5141-4be2-aa93-07a6356450c6%2Fddd0ce0a-67dc-4f15-a2ae-d669f3ff3b71%2Fay2qm6t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Estimating Car Depreciation Using Least-Squares Regression**
**Introduction:**
A statistics professor bought a new car for $35,000. For the next 5 years, she used several automotive web pages to estimate the value of the car. She then found the least-squares regression line for the data to be:
\[ \hat{y} = 35035.71 - 4142.86x \]
where \( y \) is the value of the car and \( x \) is the years since she purchased it. Predict the value of her vehicle after 6 years.
**Discussion:**
The equation provided is a linear regression model that describes the depreciation of the car's value over time. The term \( 35035.71 \) represents the initial value of the car when \( x = 0 \), which is close to the purchase price of $35,000. The slope \( -4142.86 \) indicates the annual decrease in the car's value.
**Calculation:**
To predict the value of the car after 6 years, substitute \( x = 6 \) into the regression equation:
\[ \hat{y} = 35035.71 - 4142.86 \times 6 \]
\[ \hat{y} = 35035.71 - 24857.16 \]
\[ \hat{y} = 10178.55 \]
Therefore, the predicted value of the car after 6 years is approximately $10,178.55.
This regression model can be useful for understanding how quickly assets like cars depreciate over time, allowing for better financial planning and decision-making.
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