Let's go through each question one by one.

Part (a): Finding the Equation of the Least-Squares Line

The least-squares regression line is given by the equation:

$\hat{y} = mx + b$

where:

• $m$ is the slope, calculated as $m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$
• $b$ is the y-intercept, calculated as $b = \bar{y} - m\bar{x}$

Given dataLet's go through each question one by one.

Part (a): Finding the Equation of the Least-Squares Line

The least-squares regression line is given by the equation:

$\hat{y} = mx + b$

where:

• $m$ is the slope, calculated as $m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$
• $b$ is the y-intercept, calculated as $b = \bar{y} - m\bar{x}$

Given

• Odometer readings (independent variable, $x$): 13,000, 18,000, 20,000, 25,000, 29,000, 32,000
• Retail Values (dependent variable, $y$): 52,525, 51,575, 51,300, 50,525, 49,725, 49,175

I'll calculate these values for you.

Part (b): Predicting the Retail Value for 30,000 Miles

Once we have the equation from part (a), substitute $x = 30,000$ to find the predicted retail value.

Part (c): Finding the Linear Correlation Coefficient

The linear correlation coefficient $r$ measures the strength and direction of the linear relationship between $x$ and $y$. It is given by:

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

Part (d): Interpreting the Negative Correlation

The significance of a negative correlation here would mean that as the odometer reading (miles driven) increases, the retail value of the car decreases.

I'll now perform the necessary calculations to provide specific answers for each part.Here are the solutions to each part based on the calculations:

Part (a): Equation of the Least-Squares Line

The equation of the least-squares line is: $\hat{y} = -0.17x + 54768.64$ This equation indicates that for each additional mile on the odometer, the retail value decreases by approximately $0.17.

Part (b): Predicted Retail Value for 30,000 Miles

Using the equation from part (a), the predicted retail value for a 2020 Corvette with an odometer reading of 30,000 miles is approximately: $\$49,600$

Part (c): Linear Correlation Coefficient

The linear correlation coefficient $r$ is calculated to be: $r = -0.999$ This value, being very close to -1, suggests a very strong negative linear relationship between odometer reading and retail value.

Part (d): Interpretation of the Negative Correlation

Since the correlation coefficient is negative, it indicates that as the odometer reading increases, the retail value of the car decreases. This is expected because higher mileage generally reduces a car's resale value.

Would you like further details on any of these calculations?

Here are five questions that could further explore this problem:

1. How would the predicted retail value change if the odometer reading was 40,000 miles?
2. How does the slope of the least-squares line affect predictions for higher mileage values?
3. What would be the retail value if the odometer reading was only 10,000 miles?
4. How would the correlation coefficient change if an additional point with very high mileage and low value was added?
5. How could this least-squares equation be used to estimate values for other cars of the same model year?

Tip: When interpreting the slope of a least-squares line, remember that it represents the rate of change in the dependent variable (here, retail value) for each unit increase in the independent variable (odometer reading).