Question

Regional Rent-A-Car is interested in studying how weight of a vehicle is related to the fuel efficiency for several of its rental vehicles.

A random sample of 13 vehicles from Regional Rent-A-Car yielded the data in the table.

weight (in pounds): 2000 3123 3775 2287 4000 2485 2114 2433 2525 2889 3186 3258 3622

fuel efficiency (in miles per gallon): 32.7 22.8 18.6 26.1 18.4 26.2 33.1 28.3 26.2 22.5 20.9 21.3 18.1

Part A: Identify and interpret in context the estimate of the slope for the least-squares regression line.

Part B: Identify and interpret in context the correlation coefficient.

Part C: A car weighing 2,684 pounds had an average fuel efficiency of 24.6 miles per gallon. What is this car’s residual?

Part D: Estimate the weight of a vehicle whose fuel efficiency is 20.2 miles per gallon.

Part E: Is this model the best fit for the data? Explain your reasoning. If a transformation is required, what is the equation for the new model? Show your work.
Part F: Using the transformation you gave in Part E, what would be the new estimate for fuel efficiency of the vehicle weighing 2,684 pounds? What effect did this new model have on the residual, compared to the result you found in Part C? Show all work.

Look over the rubric to see how you will be evaluated.

Answer 1

The estimate of the slope for the least-squares regression line is -0.025. The correlation coefficient is -0.694. To find the residual for a car weighing 2,684 pounds with an average fuel efficiency of 24.6 miles per gallon, we use the equation of the least-squares regression line. The predicted value is 28.286 miles per gallon, so the residual is 24.6 - 28.286 = -3.686 miles per gallon.

Part A:

The estimate of the slope for the least-squares regression line is -0.025.

Part B:

The correlation coefficient (r) can be calculated using the formula:

ρ (X,Y) = cov (X,Y) / σX.σY

The correlation coefficient is -0.694.

Part C:

To find the residual for a car weighing 2,684 pounds with an average fuel efficiency of 24.6 miles per gallon, we use the equation of the least-squares regression line. The predicted value is 28.286 miles per gallon, so the residual is 24.6 - 28.286 = -3.686 miles per gallon.

Part D:

To estimate the weight of a vehicle with a fuel efficiency of 20.2 miles per gallon, we use the equation of the least-squares regression line. The predicted weight is 3480 pounds.

Part E:

Based on the scatter plot and the correlation coefficient, the line seems to fit the data moderately well. The correlation coefficient of -0.694 indicates a negative relationship between fuel efficiency and weight, which means that as weight increases, fuel efficiency tends to decrease. However, the scatter plot shows some deviations from the line, suggesting that there may be other factors influencing fuel efficiency.

Part F:

Using the new model derived from the natural logarithm transformation of weight, the predicted fuel efficiency for a vehicle weighing 2,684 pounds is 24.727 miles per gallon. Compared to the previous model, this new model slightly reduced the residual from -3.686 to -3.473 miles per gallon.