https://file.aiquickdraw.com/mathbot/file/1730548096590nl09qmkz.jpg

To solve this problem, we need to follow these steps:

### Step-by-Step Solution:
1. **Define Variables**:
   - Let $ x $ represent the odometer reading.
   - Let $ y $ represent the retail value of the car.

2. **Construct the Regression Line Equation** (for part a):
   - We use the form $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept.
   - Calculate $ m $ (slope) using the formula:
     \[
     m = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}
     \]
   - Calculate $ b $ (y-intercept) using:
     \[
     b = \bar{y} - m\bar{x}
     \]
   - Round constants to the nearest thousandth.

3. **Prediction (part b)**:
   - Substitute $ x = 30,000 $ into the equation from part (a) to predict the retail price.

4. **Linear Correlation Coefficient (part c)**:
   - Use the formula for the correlation coefficient $ r $:
     \[
     r = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2}}
     \]

Would you like me to go through each calculation step-by-step or focus on any specific part? 

### Additional Questions:
1. How do you find the mean ($ \bar{x} $ and $ \bar{y} $) for a data set?
2. What does the slope in the regression line represent in this context?
3. Why is the correlation coefficient important in data analysis?
4. How does changing one value in the data affect the regression line?
5. What would a correlation coefficient close to 1 or -1 indicate?

**Tip:** Understanding how to interpret the slope and intercept in a real-world context helps in making predictions and decisions based on the data.

The following table gives the retail values of a 2004 Corvette Z06 for various odometer readings:

| Odometer Reading (x) | Retail Value (y) |
|----------------------|------------------|
| 13,000               | $46,100         |
| 18,000               | $44,600         |
| 20,000               | $43,300         |
| 25,000               | $41,975         |
| 29,000               | $40,975         |
| 32,000               | $39,750         |

Questions:
1. Find the equation of the regression line for the data. Round constants to the nearest thousandth.
2. Use the equation from part a to predict the retail price of such a car with an odometer reading of 30,000.
3. Find the linear correlation coefficient for these data.

Learn how to perform linear regression analysis on the retail values of a 2004 Corvette Z06 based on various odometer readings. This solution includes calculating the regression line equation, predicting retail values for specific readings, and finding the linear correlation coefficient. Ideal for high school statistics students.

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