Question

the fare charged for a rideshare service is a function of the distance traveled. however, the fare differs according to the time of day, availability, and other variables. the distance and fares for 10 rides are shown in the table. the equation of the least - squares regression line is \\(\hat{y}=5.21 + 2.33x\\), where \\(\hat{y}\\) is the predicted fare and \\(x\\) is the distance. what is the residual for the rideshare cost with a distance of 5 miles? \\(\circ - 0.34\\) \\(\circ 0.34\\) \\(\circ 2.33\\) \\(\circ 5.21\\) \\(\

$$\begin{array}{|c|c|}\\hline \\text{distance (miles)}&\\text{fare (dollars)}\\\\ \\hline 1&1.91\\\\ \\hline 3&13.68\\\\ \\hline 5&16.52\\\\ \\hline 8&24.15\\\\ \\hline 10&24.79\\\\ \\hline 12&39.87\\\\ \\hline 15&40.24\\\\ \\hline 16&53.84\\\\ \\hline \\end{array}$$

\\)

Explanation:

Step1: Recall residual formula

Residual is calculated as the actual value minus the predicted value, i.e., \( \text{Residual} = y - \hat{y} \).

Step2: Find predicted fare for \( x = 5 \)

Given the regression equation \( \hat{y} = 5.21 + 2.33x \). Substitute \( x = 5 \) into the equation:
\( \hat{y} = 5.21 + 2.33\times5 \)
\( \hat{y} = 5.21 + 11.65 = 16.86 \)

Step3: Find actual fare for \( x = 5 \)

From the table, when distance \( x = 5 \) miles, the actual fare \( y = 16.52 \) dollars.

Step4: Calculate the residual

Using the residual formula: \( \text{Residual} = y - \hat{y} = 16.52 - 16.86 = -0.34 \)

Answer:

-0.34