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Let's break down the problem in steps:

1. **Identify the Data**: The table provides retail prices for a car from 1998 to 2002, with `x = 1` representing the year 1998, and `x` increments by 1 for each subsequent year.
   
   - **Data points**:
     - $ (x = 1, y = 15375) $
     - $ (x = 2, y = 11554) $
     - $ (x = 3, y = 10567) $
     - $ (x = 4, y = 9916) $
     - $ (x = 5, y = 8883) $

2. **Find the Regression Equation**:
   To find the regression equation $ y = mx + b $ (a linear fit), we would calculate:
   - **Slope (m)** and **intercept (b)** using the least squares method, based on the points above.

3. **Predict for 2004**: Since 1998 is represented by $ x = 1 $, the year 2004 corresponds to $ x = 7 $.

4. **Determine Year for Value of $2486**: Set $ y = 2486 $ in the regression equation and solve for $ x $.

I'll calculate the regression equation now.The regression equation, based on the data, is:

\[
y = -1462.2x + 15645.6
\]

Using this equation:

1. **Prediction for 2004**: In 2004 ($ x = 7 $), the average retail price of the car is estimated to be approximately **$5410.20**.

2. **Year when the car's value will reach $2486**: Solving for $ x $ when $ y = 2486 $ gives an approximate value of $ x \approx 9 $. This corresponds to the year **2006**.

Would you like further details on these calculations, or do you have other questions?

### Related Questions:
1. How do we calculate the slope and intercept of a regression line?
2. Why does the linear model assume a constant rate of depreciation?
3. What other methods could predict prices if the data isn't linear?
4. How does the correlation coefficient ($ r $) relate to the accuracy of predictions?
5. What happens if we need to predict prices far outside the given data range?

### Tip:
When using a regression equation to predict values outside the given data range, be cautious of "extrapolation," as the model may not hold accurately far beyond the observed data points.

Use a calculator to find the regression equation. Predict the average retail price of a car in 2004. Determine the year when the car will be worth $2486.

Learn how to calculate a regression equation to predict car prices over time. This example predicts the average retail price in 2004 and determines the year when a car's value will reach $2486, using given data from 1998 to 2002.

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