Exercise 30 in Section 3.4 gave data on x testing temperature and y dynamic shear modulus for a particular asphalt binder type. A scatterplot of x and y log(y) shows a substantial linear pattern, suggesting that these variables are related by the simple linear regression model.
a. What probabilistic model for relating y dynamic shear modulus to x testing temperature is implied by the simple linear regression relationship between x and y9?
b. Summary quantities calculated from the data are
Calculate estimates of the parameters for the model in part (a), and then obtain a point prediction of dynamic shear modulus when temperature is 35°F.
In the article “Sensitivity of Oklahoma Binders on Dynamic Modulus of Asphalt Mixes and Distress Functions” (J. Mater. Civ. Engr., 2012: 1076–1088), researchers measured various physical characteristics of performance grade asphalt binders commonly used in Oklahoma. One important physical characteristic is dynamic shear modulus, G (kPa), which is the ratio of maximum shear stress to the maximum shear strain and is a measure of the stiffness or resistance of the asphalt binder to deformation under load. In one experiment, the researchers measured the dynamic shear modulus of the asphalt binder samples over a range of testing temperatures (°C). The following is the corresponding data for binder type PG64-22:
a. Construct a scatterplot of y dynamic shear modulus versus temperature. Would it be reasonable to characterize the relationship between the two variables as approximately linear?
b. Transform only the dependent variable y so that a scatterplot of the transformed data shows a substantial linear pattern. Then fit a straight line to this data, use the line to establish an approximate relationship between x and y, and predict the dynamic shear modulus when the temperature is 35°C.
c. Plot the residuals from your linear fit in part (b) and look for any patterns that might suggest an inappropriate choice of transformation. If necessary, return to part (b) and try a different transformation.