An Analyst Has Identified 3 Independent Variables (X₁, X₂, X₃)

Multiple Choice

An analyst has identified 3 independent variables (X₁, X₂, X₃) which might be used to predict Y. He has computed the regression equations using all combinations of the variables and the results are summarized in the following table. Why is the R² value for the X₃ model the same as the R² value for the X₁ and X₃ model, but the Adjusted R² values differ? $\begin{array}{lcccl}\text { Independent}\\\text { Variable in the}&\text { Adjusted}\\ \hline\text { Model } & \mathrm{R}^{2} & -\mathrm{R}^{2} & \mathrm{~S}_{\mathrm{e}} & \text { Parameter Estimates } \\\hline \mathrm{X}_{1} & 0.00089 & -0.124 & 23.548 & \mathrm{~b}_{0}=93.7174, \mathrm{~b}_{1}=0.922 \\\mathrm{X}_{2} & 0.3870 & 0.3104 & 18.448 & \mathrm{~b}_{0}=57.0803, \mathrm{~b}_{2}=1.545 \\\mathrm{X}_{1} \text { and } \mathrm{X}_{2} & 0.3910 & 0.2170 & 19.654 & \mathrm{~b}_{0}=50.2927, \mathrm{~b}_{1}=1.952, \mathrm{~b}_{2}=1.554 \\\mathrm{X}_{3} & 0.8413 & 0.8214 & 9.3858 & \mathrm{~b}_{0}=31.6238, \mathrm{~b}_{3}=1.132 \\\mathrm{X}_{1} \text { and } \mathrm{X}_{3} & 0.8413 & 0.7960 & 10.033 & \mathrm{~b}_{0}=31.133, \mathrm{~b}_{1}=0.148, \mathrm{~b}_{3}=1.132 \\\mathrm{X}_{2} \text { and } \mathrm{X}_{3} & 0.9863 & 0.9824 & 2.948 & \mathrm{~b}_{0}=14.169, \mathrm{~b}_{2}=0.985, \mathrm{~b}_{3}=0.995 \\\text { All three } & 0.9871 & 0.9807 & 3.085 & \mathrm{~b}_{0}=11.113, \mathrm{~b}_{1}=0.899, \mathrm{~b}_{2}=0.990 \\&&&&\mathrm{~b}_{3}=0.993\end{array}$

A) The standard error for X₁ is greater than the standard error for X₃.
B) X₁ does not reduce ESS enough to compensate for its addition to the model.
C) X₁ does not reduce TSS enough to compensate for its addition to the model.
D) X₁ and X₃ represent similar factors so multicollinearity exists.