Steel microfibers are an alternative to conventional rebar reinforcement of concrete structures. The article “Measurement of Average Tensile Force for Individual Steel Fiber Using New Direct Tension Test” (J. Test. Eval. 2016: 2403–2413) proposes a new evaluation method for concrete specimens infused with twisted steel micro rebar (TSMR) fibers, which were loaded until a crack occurred. The accompanying data on y = load until cracking (lb), x₁ = diameter at break (in), x₂ = number of TSMR fibers per in², and x₃ = concrete compressive strength (psi) appears in the article.

a. Determine the estimated regression equation for this data.
b. Perform variable utility tests on each of the three explanatory variables. Can each one be judged useful given that the other two are included in the model?
c. Calculate R² and adjusted R² for this three-predictor model.
d. Perform a multiple regression of y on just x₂ and x₃. Determine both R² and adjusted R² for this reduced model. How do they compare to the values in part (c)? Explain.

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y X1 X2 X3 213 2.778 1.2 7020 706 2.725 3.3 7020 180 989 2.686 6.0 7960 326 2.821 4.0 y X1 688 2.735 7960 7090 7090 440 2.683 3.4 7020 1046 2.755 984 2.835 8.2 5680 272 2.734 2.725 155 2.779 2.3 5680 1168 251 2.712 2.3 5680 821 311 2.712 1.6 7960 2.750 6.1 418 2.740 4.4 1102 56 91 97 479 2.810 2.7 324 2.845 1.9 404 2.755 3.0 7090 X2 X3 3.4 6130 6130 2.705 2.6 7.7 6130 3.2 6880 7.4 7270 7270 7270 2.708 9.2 2.860 1.6 3.002 1.3 2.749 1.9 7390 2950 2950 2950