An experiment was conducted regarding a quantitative analysis of factors found in high-density lipoprotein (HDL) in a sample of human blood serum. Three variables thought to be predictive of, or associated with, HDL measurement (Y) were the total cholesterol (X1) and total triglyceride (X2) concentrations in the sample, plus the presence or absence of a certain sticky component of the serum called sinking pre-beta or SPB (X3), coded as 0 if absent and 1 if present. The data obtained are shown in the following table.
a. Test whether X1, X2, or X3 alone significantly helps to predict Y.
b. Test whether X1, X2, and X3 taken together significantly help to predict Y.

c. Test whether the true coefficients of the product terms X1X3 and X2X3 are simultaneously zero in the model containing X1, X2, and X3 plus these product terms. Specify the two models being compared, and state the null hypothesis in terms of regression coefficients. If this test is not rejected, what can you conclude about the relationship of Y to X1 and X2 when X3 equals 1 compared to the same relationship when X3 equals 0?
d. Test (at α = .05) whether X3 is associated with Y, after taking into account the combined contribution of X1 and X2. What does your result, together with your answer to part (c), tell you about the relationship of Y with X1 and X2 when SPB is present compared to the same relationship when it is absent?
e. What overall conclusion can you draw about the association of Y with the three independent variables for this data set? Specify the two models being compared, and state the appropriate null hypothesis in terms of regression coefficients.