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Based on data taken from airline fares and distances flown, it is determined that the equation of the least-squares regression line is [tex]\hat{y} = 102.50 + 0.65x[/tex], where [tex]\hat{y}[/tex] is the predicted fare and [tex]x[/tex] 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?

A. 102.50
B. 312.50
C. 427.50
D. 542.50



Answer :

GinnyAnswer
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.

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