Question

question 2
1 pts
there is a strong linear relationship between the distance traveled (in kilometers) and the fare (in dollars) for those who utilize rideshare services such as uber. the regression equation to predict fare based on distance traveled is:
predicted fare = 3.16 + 1.51 (distance traveled)
which one of the following statements is a correct interpretation of this equation?
○ for each additional dollar spent on a rideshare service, the distance traveled increases by 1.51 kilometers.
○ as the distance traveled increases by one kilometer, the predicted fare increases by $3.16.
○ the fare for the rideshare service is predicted to go up by $1.51 with each additional kilometer traveled.
○ the predicted fare for a one - kilometer trip is $1.51.
○ the regression equation is not valid because the slope can be no less than - 1 and no greater than 1.

Explanation:

Step1: Recall Regression Equation

The regression equation is \( \text{Predicted fare} = 3.16 + 1.51 (\text{distance traveled}) \). In a linear regression \( \hat{y} = b_0 + b_1 x \), \( b_0 \) is the intercept (value when \( x = 0 \)) and \( b_1 \) is the slope (change in \( \hat{y} \) per unit change in \( x \)). Here, \( x \) is distance traveled (km), \( \hat{y} \) is predicted fare (\$), \( b_0 = 3.16 \), \( b_1 = 1.51 \).

Step2: Analyze Each Option

• Option 1: Swaps independent (distance) and dependent (fare) variables. The equation predicts fare from distance, not distance from fare. Eliminate.
• Option 2: The slope \( b_1 = 1.51 \), not 3.16. The intercept 3.16 is fare when distance = 0. Eliminate.
• Option 3: Slope \( b_1 = 1.51 \) means for each additional km (unit change in \( x \)), predicted fare (\( \hat{y} \)) increases by \$1.51. This matches the slope interpretation.
• Option 4: When distance = 1, predicted fare is \( 3.16 + 1.51(1) = 4.67 \), not \$1.51. Eliminate.
• Option 5: Slope can be any real number, no restriction to \([-1, 1]\). Eliminate.

Answer:

The fare for the rideshare service is predicted to go up by \$1.51 with each additional kilometer traveled.