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.

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x, 0 0 0 0 0 0 0 0 1 1 1 1 5623666 95227 681 18 3 5 8 4 9 2 3 2 802 322233223 22322 286824 8648 Y-34 41 42 39 2 3 3 5 4 5 3 0 6 US$ 68 92 56 210 996 93 15 400 103 02 130 49 9141508293 6276849462185 131222132 322 4566 67 5 524 45.57 5 00000000000111 2-158 3.6 37 7 6 74957 X. | 2 5 6 66142 139640 121211112 x, 28 2 532 17 360 37 201 997 29 5 1 8 0 699658 1 11222 3 2 2 2 2 Y 78794489529 702 545-5 4 4 4 4 Y Regressed on X2 DF Sum of Squares Mean Square F ValuePr> F Source Model Error Corrected Total 21.339709 4592.279339 4813.619048 21.339709 0.19 0.6687 40 114806983 41 R-Square | CoefVar | Root MSE Y Mean 0.004625 22.43378 10.71480 47.76190 Source DF TypeSS Mean Square F Value Pr> F X2 121.33970871 21.33970871 0.19 0.6687 Y Regressed on X3 DF Type I SS Mean Square F Value Pr> F X3 735.2054113 735 2054113 7.58 0.0088 Y Regressed on X1, X2, X3, X1X2, and X2X3 Source DF Type ISS Mean Square FValue Pr> F X1 X2 X3 X1x3 X2x3 46.2355575 89.1465681 684 3651973 48.6904324 1 26.1296111 46.2355575 0.45 0.5078 89.1465681 0.88 0.3591 684.3651973 6,62 48.6904324 047 0.4968 26,129811025 0181