The first four columns of the following data give the average precipitation (inches averaged over 30 years) in April and May for five western U. S. cities and five eastern U. S. cities. (Source: 1993 Almanac and Book of Facts. Pharos Books, Scripps Howard Company, New York.) The last three columns include numbers we use later in the exercise.
(a) Plot May precipitation versus April using E and Was plot symbols to represent the coasts. What do you conclude from the plot? Is it appropriate to fit a single straight line for both coasts?
(b) Regress the May precipitation on the April precipitation for each region. Add together the error sums of squares and refer to this as the full model residual sum of squares where the full model allows two different slopes and two different intercepts. Compute the difference in the two slopes and in the two intercepts.
(c) Now, regress the May precipitation on the April precipitation using all n = 10 points. The error sum of squares here is the reduced model residual sum of squares. The reduced model forces the same intercept and slope for the two groups. Compare the full to the reduced model using an F-test. What degrees of freedom did you use?
(d) Run a multiple regression of May precipitation on columns SE, XE, and XW. What do the coefficients on XE and XW represent? Have you seen these numbers before? How about the error sum of squares and the coefficient on SE? Write out the X matrix for this regression. What would happen to the rank of X if we appended the column of 10 April precipitation numbers to it?
(e) Finally run a multiple regression of May precipitation on April precipitation, SE, and XE. Write out the X matrix for this regression. Compute the F-test for the hypothesis that SE and XE can be omitted from this model. Have you seen this test before? The coefficient on XE
in this regression estimates the difference of the two slopes in (b) and thus can be used to test the hypothesis of parallel lines. Test the hypothesis that the lines have equal slopes. Omission of SE from this model produces two lines emanating from the same origin. Test the hypothesis that both lines have the same intercept (with possibly different slopes).