the estimated value of the car after 8 years is $6,074.80.
To calculate the value of the car after 8 years, we perform a linear regression analysis on the given data points. Here's a step-wise breakdown of the calculations:
1. Data Points: We have the following data points from the table:
- Year 0 (x=0), Value $20,000 (y=20,000)
- Year 1 (x=1), Value $18,000 (y=18,000)
- Year 2 (x=2), Value $16,200 (y=16,200)
- Year 3 (x=3), Value $14,580 (y=14,580)
- Year 4 (x=4), Value $13,122 (y=13,122)
2. Linear Regression: We then calculate the slope and intercept of the line that best fits this data using linear regression. The formula for a line is y = mx + b, where m is the slope and b is the intercept.
3. Slope (m): The slope of the regression line is calculated to be approximately \(-1717.6\). This means that each year, the value of the car decreases by about $1,717.60.
4. Intercept (b): The intercept of the line, representing the initial value of the car when x=0 (at year 0), is approximately $19,815.60.
5. Determination Coefficient (r²): The r-squared value is \(0.9961\), which is very close to 1. This indicates a very good fit of the regression line to the data points.
6. Standard Error: The standard error of the estimate is approximately \(61.72\). This gives us a measure of the accuracy of the predictions.
7. Prediction: Using the regression equation y = -1717.6x + 19815.6, we predict the value of the car at year 8.
8. Predicted Value: Substituting x = 8 into the equation gives us the predicted value of the car, which is approximately $6,074.80.
Therefore, the estimated value of the car after 8 years is $6,074.80.