Let's break down the problem in the image step by step:

(a) Interpretation of the regression equation

The regression line equation is given as:

$y = 8.89 + 7.73x$

Where:

• $y$ is the mileage (in thousands of miles),
• $x$ is the age of the car (in years).

Practical interpretation:

• 8.89: This represents the y-intercept. When $x = 0$ (i.e., for a brand new car), the predicted mileage is 8.89 thousand miles. In reality, this would imply that a new car might already have some mileage, likely due to factory testing or delivery miles.

• 7.73: This is the slope of the line. It indicates that for every additional year of the car's age, the mileage increases by 7.73 thousand miles. So, older cars tend to have accumulated more mileage over time.

(b) Predicting mileage for a 5-year-old and 8-year-old car

We will substitute $x = 5$ and $x = 8$ into the regression equation $y = 8.89 + 7.73x$ to find the predicted mileage.

1. For a 5-year-old car: $y = 8.89 + 7.73 \times 5 = 8.89 + 38.65 = 47.54 \text{ thousand miles}$

2. For an 8-year-old car: $y = 8.89 + 7.73 \times 8 = 8.89 + 61.84 = 70.73 \text{ thousand miles}$

Comment on the suitability:

• 5-year-old car: The predicted mileage for a 5-year-old car is 47.54 thousand miles. Looking at the table, the actual mileage for a 5-year-old car is 49 thousand miles. The predicted value is quite close to the actual value, indicating that the regression line provides a reasonable prediction for a 5-year-old car.

• 8-year-old car: The regression predicts a mileage of 70.73 thousand miles for an 8-year-old car. However, the oldest car in the data set is only 6.5 years old, with a mileage of 58 thousand miles. Since 8 years is beyond the range of the provided data, the prediction is based on an extrapolation, which may not be as reliable as interpolation within the range of observed data. Therefore, this prediction should be taken with caution.

Do you want more details or have any further questions?
Here are 5 related questions to deepen understanding:

1. How would the predictions change if the slope were higher, say 9 instead of 7.73?
2. What does it mean when we extrapolate beyond the given data range, and what are the risks?
3. How would you test if this regression model is the best fit for the data?
4. If a car has been used more intensively, how might its mileage deviate from the regression line?
5. How would the interpretation change if the intercept was a negative number?

Tip: Extrapolating beyond the observed data can lead to inaccurate predictions, as the relationship might change outside the given data range. Always interpret with caution.