The data2 in Temp.Long.Lat.txt give the average (over the years 1931 to 1960) daily minimum January temperature in degrees Fahrenheit with the latitude and longitude of 56 US cities. Let Y, X1, X2 denote the temperature and the centered latitude and longitude variables, respectively.

(a) Using R commands to fit the following models, report the least squares estimates of the regression parameters, the estimated intrinsic error variance, as well as R2 and R2(adj) for each of the models:

(1) the basic multiple regression model Y = β0 + β1X1 + β2X2 + ε,

(2) the model Y = β0+β1X1+β2X2+β3X1X2+ε, which adds the interaction term to the basic model, and

(3) the model Y = β0 +β1X1 +β1, 2X2 1 +β2X2 +β2, 2X2 2 +β3X1X2 +ε, which includes second-degree polynomial terms in X1 and X2 and their interaction.

(Note that indexing coefficients of polynomial terms by double subscripts, as done above, is not very common.)

(b) Use each model’s estimated intrinsic variance and the corresponding degrees of freedom to find each model’s SSE and (their common) SST.

(c) The average minimum January temperature in Mobile,AL, and its latitude and longitude are 44, 31.2, 88.5. For each of the three models, give the fitted value and the residual for Mobile, AL.