United Oil Company is attempting to develop a reasonably priced unleaded gasoline that will deliver higher gasoline mileages than can be achieved by its current unleaded gasolines. As part of its development process, United Oil wishes to study the effect of two independent variables— x1 , amount of gasoline additive RST (0, 1, or 2 units), and x2, amount of gasoline additive XST (0, 1, 2, or 3 units), on gasoline mileage, y. Mileage tests are carried out using equipment that simulates driving under prescribed conditions. The combinations of x1 and x2 used in the experiment, along with the corresponding values of y, are given in the page margin.
a. Discuss why the data plots given in the page margin indicate that the model y = β0 + β1x1 + β2x21 + β3x2 + β4x22 + ε might appropriately relate y to x1 and x2. If we fit this model to the data in the page margin, we find that the least squares point estimates of the model parameters and their associated p-values (given in parentheses) are b0 28.1589 (<.001), b1 = 3.3133 (<.001), b2 = – 1.4111 (<.001), b3 = 5.2752 (<.001), and b4 = – 1.3964 (<.001). Moreover, consider the mean mileage obtained by all gallons of the gasoline when it is made with one unit of RST and two units of XST (a combination that the data in the page margin indicates would maximize mean mileage). A point estimate of and a 95 percent confidence interval for this mean mileage are 35.0261 and [34.4997, 35.5525]. Using the above model, show how the point estimate is calculated.
b. If we add the independent variable x1x2 to the model in part a, we find that the p- value related to x1x2 is .9777. What does this imply?