An independent mail delivery service wants to study factors that affect the daily gas usage of its delivery trucks. Using data collected from different trucks on various days, a company analyst uses software to fit a regression model of the following form. \[ \hat{y}=76.8+5.4 x_{1}-4 x_{2}+4 x_{3}+0.04 x_{4} \] In this model are the following variables. \[ \begin{array}{l} y=\text { volume of gasoline used (in gallons) } \\ x_{1}=\text { weight of truck (in tons) } \\ x_{2}=\text { tire pressure (in psi, pounds per square inch) } \\ x_{3}=\text { weight of initial package load (in hundreds of pounds) } \\ x_{4}=\text { total distance driven while delivering packages (in miles) } \end{array} \] Answer the questions below for the interpretation of the coefficient of $x_{1}$ in this model. (a) Holding the other variables fixed, what is the average change in daily fuel used for each additional ton that a truck weighs? gallon(s) (b) Is this change an increase or a decrease? increase decrease
Solution
Step 1 ：The question is asking for the interpretation of the coefficient of $x_{1}$ in the given regression model. The coefficient of $x_{1}$ is 5.4. This coefficient represents the average change in the dependent variable (in this case, the volume of gasoline used) for each unit increase in the independent variable $x_{1}$ (in this case, the weight of the truck), holding all other variables constant.
Step 2 ：In this case, the unit of $x_{1}$ is tons, so the coefficient represents the average change in the volume of gasoline used for each additional ton that a truck weighs, holding all other variables constant.
Step 3 ：Since the coefficient is positive, this means that as the weight of the truck increases by one ton, the volume of gasoline used is expected to increase, on average, by 5.4 gallons, assuming all other variables remain constant.
Step 4 ：Therefore, the answer to part (a) is \(\boxed{5.4}\) gallons, and the answer to part (b) is increase.
