Question

a frequent flyer was interested in the relationship between dollars spent on flying and the distance flown. she sampled 20 frequent flyers of a certain airline. she collected the number of miles flown in the previous year and the total amount of money the flyer spent. a regression line of distance flown on money spent was fit to the data, and the intercept and slope were calculated to be a = 24,000 and b = 10. one of the randomly sampled frequent flyer was found to have spent $2500 and flown 57,000 miles. the residual for this observation is: select all that apply: a. the predicted distance flown for someone who spent $2500 is 49,000 miles. b. the actual distance flown (57,000 miles) is greater than the predicted distance. c. the residual for this observation is -8000 miles. d. the residual for this observation is +8000 miles. e. the regression line predicts that someone who spent $0 would have flown 24,000 miles.

Explanation:

Step1: Recall regression - line formula

The regression - line equation is $\hat{y}=a + bx$, where $a$ is the intercept, $b$ is the slope, $x$ is the independent variable (money spent), and $\hat{y}$ is the predicted value of the dependent variable (distance flown). Given $a = 24000$ and $b = 10$, and $x = 2500$.

Step2: Calculate the predicted distance

Substitute the values into the formula: $\hat{y}=24000+10\times2500=24000 + 25000=49000$ miles.

Step3: Recall the definition of residual

The residual $e=y-\hat{y}$, where $y$ is the actual value and $\hat{y}$ is the predicted value. Given $y = 57000$ miles and $\hat{y}=49000$ miles.

Step4: Calculate the residual

$e=57000 - 49000=8000$ miles. Also, when $x = 0$, $\hat{y}=a=24000$ miles.

Answer:

A. The predicted distance flown for someone who spent $2500 is 49,000 miles.
B. The actual distance flown (57,000 miles) is greater than the predicted distance.
D. The residual for this observation is +8000 miles.
E. The regression line predicts that someone who spent $0 would have flown 24,000 miles.