Q. 4.65 - Elementary Statistics

Question

Corvette Prices. In Exercise 4.59. you determined a regression equation that can be used to predict the price of a Corvette, given its age.

a. Should that regression equation be used to predict the price of a 4-year-old Corvette? a 10-year-old Corvette? Explain your answers.

b. For which ages is the use of the regression equation to predict price reasonable?

Step-by-Step Solution

Answer

a. Using the regression equation for a 10 -year-old Corvette is not applicable (and would be extrapolation) since that age lies outside the range of ages included in the sample data.

b. The age between $1$ and $6$ years is the use of the regression equation to predict price reasonable.

Part (a) Step 1: Given Information

The following table illustrates the values of $10$ randomly selected Corvettes, divided by their age and price expressed in hundreds of dollars. The age is expressed by $X$, in the form of years, and price by $Y$, in the form of hundreds of dollars.

Part (a) Step 2: Explanation

Here is the regression equation to determine the price $\left(\mathrm{y}\right)$ from the age $\left(x\right).$

$\hat{y}=456.6019-27.9029x$

Applying the concept:

Concept Used:

A regression line can be used to predict a value of the predictor variable if the value falls within the range of $x-$values given

The age of four years falls within the range of the provided $x-$values. As a result, using the derived regression line to estimate the price is plausible.

The age of ten falls beyond the range of the specified x-values. As a result, using the generated regression line to predict the price is not a good idea.

Part (b) Step 3: Given Information

To determine the age interval over which the regression equation can be utilized to create a price prediction. 

Part (b) Step 4: Explanation

From the given ages,

The minimum is $1-$year-old.

The maximum is $6-$years-old.

Using the calculated regression equation, it is reasonable to predict the price based on a Corvette that is between $1$and $6$ years old.