To determine the fare for the flight based on the given data, we need to follow these steps:
1. Identify the Given Information:
- The regression equation [tex]\(\hat{y} = 102.50 + 0.65x\)[/tex]
- Distance [tex]\(x = 500\)[/tex] miles
- Residual [tex]\(e = 115.00\)[/tex]
2. Calculate the Predicted Fare:
- The predicted fare [tex]\(\hat{y}\)[/tex] can be found by substituting the distance [tex]\(x = 500\)[/tex] into the regression equation.
[tex]\[
\hat{y} = 102.50 + 0.65 \times 500
\][/tex]
- Calculate the value:
[tex]\[
\hat{y} = 102.50 + 325.00 = 427.50
\][/tex]
3. Calculate the Actual Fare:
- The residual [tex]\(e\)[/tex] is the difference between the actual fare [tex]\(y\)[/tex] and the predicted fare [tex]\(\hat{y}\)[/tex]:
[tex]\[
e = y - \hat{y}
\][/tex]
- Rearrange to find the actual fare [tex]\(y\)[/tex]:
[tex]\[
y = \hat{y} + e
\][/tex]
- Substitute the predicted fare and residual:
[tex]\[
y = 427.50 + 115.00 = 542.50
\][/tex]
Conclusion:
The fare for the flight was 542.50.
