Question

A major expense for car drivers is fuel, so it is of interest to compare fuel economy across different vehicles. The data below represents the fuel economy (in kilometres per litre) and weight (in tonnes) of 20 different vehicles. Weight in tonnes 1.6, 1.6, 1.12, 1.29, 1.58, 1.63, 1.51, 1.51, 1.53, 1.56, 1.87, 1.83, 1.11, 1.7, 1.9, 1.28, 1.23, 1.02, 1.13, 1.09 Fuel Efficiency in kilometres per litre 8.92, 10.86, 13.59, 11.63, 8.57, 8.87, 13.35, 11.3, 9.62, 8.71, 8.29, 5.61, 12.38, 8.26, 10.84, 14.33, 12.47, 11.62, 15.03, 15.17 The linear regression model is given by Y = Bo+B1X + ε where X is the predictor and Y is the response variable.

a) Estimate the values of Bo and B1 number (Enter your answer correct to 2 decimal places) 61 number (Enter your answer correct to 2 decimal places)
b) What proportion of variability in the response is explained by the predictor? number (Enter your answer as a proportion, correct to 3 decimal places)
c) Determine the observed sample correlation coefficient between the predictor and the response variable. number (Enter your answer correct to 2 decimal places)
d) Perform a hypothesis test to determine whether the fuel economy is related to vehicle weight.

Answer 1

The slope of the least-squares line relates to the change in fuel efficiency with vehicle weight. Predicting fuel efficiency for weights well beyond the original data range is unreliable. The correlation and R-squared value suggest a moderate negative relationship where vehicle weight accounts for approximately 31% of the variability in fuel efficiency.

Step-by-step explanation:

Regarding the practical interpretation of the slope of the least-squares line in terms of fuel efficiency and weight, it represents the change in fuel efficiency for each one-tonne increase in vehicle weight. If the slope is negative, it suggests that heavier vehicles tend to have lower fuel efficiency, and vice versa.

For a car that weighs 4,000 pounds (approximately 1.814 tonnes), we would use the provided regression model to predict its fuel efficiency. However, as the specific equation is not given, an exact answer cannot be provided without the values of B0 (the y-intercept) and B1 (the slope).

Predicting the fuel efficiency of a car weighing 10,000 pounds using the least-squares line may not be advisable. As extrapolating far beyond the range of the provided data can lead to inaccurate predictions, we must exercise caution in extending the regression model to weights not within the original data set.

In terms of model fit and significance, we analyze the provided data for the correlation and the coefficient of determination (R-squared value). A correlation of -0.56 suggests a moderate negative relationship, and with an R-squared value of approximately 31%, we can conclude that roughly a third of the variability in fuel efficiency is explained by vehicle weight.

Lastly, to establish if there are any outliers in the data, one would typically look for points that deviate significantly from the pattern seen in the scatter plot. Without specific values, we can not identify outliers in this case.