Question

based on data taken from airline fares and distances flown, it is determined that the equation of the least-squares regression line is \\(\hat{y} = 102.50 + 0.65x\\), where \\(\hat{y}\\) is the predicted fare and \\(x\\) is the distance, in miles. one of the flights was 500 miles and its residual was 115.00. what was the fare for this flight? \\(\bigcirc\\) 102.50 \\(\bigcirc\\) 312.50 \\(\bigcirc\\) 427.50 \\(\bigcirc\\) 542.50

Explanation:

Step1: Recall the formula for residual

The residual is calculated as the actual value minus the predicted value, i.e., \( \text{Residual} = y - \hat{y} \), where \( y \) is the actual fare (what we need to find), and \( \hat{y} \) is the predicted fare.

Step2: Calculate the predicted fare \( \hat{y} \)

Given the regression equation \( \hat{y} = 102.50 + 0.65x \) and \( x = 500 \) miles. Substitute \( x = 500 \) into the equation:
\( \hat{y} = 102.50 + 0.65 \times 500 \)
\( \hat{y} = 102.50 + 325 \)
\( \hat{y} = 427.50 \)

Step3: Solve for the actual fare \( y \)

We know the residual is \( 115.00 \), and from the residual formula \( \text{Residual} = y - \hat{y} \), we can rearrange it to \( y = \text{Residual} + \hat{y} \).
Substitute the known values: \( y = 115.00 + 427.50 \)
\( y = 542.50 \)

Answer:

542.50