Researchers carried out a study to see how y = ultimate deflection (mm) of reinforced ultrahigh toughness cementitious composite beams were influenced by x1 = shear span ratio and x2 = splitting tensile strength (MPa), resulting in the accompanying data ("Shear Behavior of Reinforced Ultrahigh Toughness Cementitious Composite Beams without Transverse Reinforcement," J. of Materials in Civil Engr., 2012: 1283-1294):

a. Here is Minitab output from fitting the model with predictors x1, x2, and x3 = x1x2:
The regression equation is
y = 17.3 - 6.37 x1 - 3.66 x2 + 1.71 x1x2

Carry out a test of model utility.
b. Should the interaction predictor be retained in the model? Carry out a test of hypotheses using a significance level of .05.
c. The estimated standard deviation of Y^ when x1 = 3 and x2 = 6 is sY^ = .555. Calculate and interpret a confidence interval with a 95% confidence level for true average deflection under these circumstances.
d. Using the information in (c), calculate and interpret a prediction interval using a 95% confidence level for a future value of ultimate deflection to be observed when x1 = 3 and x2 = 6.

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2.04 3.55 3.11 2.04 6.073.26 3.06 3.06 6.07 10.25 4.08 3.553.11 4.08 6.16 13.48 2.06 3.623.94 2.06 6.163.53 08 3.62 3.36 .08 5.89 6.49 4.11 3.622.72 4.11 5.89 12.48 2.01 6.18 2.82 3.02 6.18 .03 6.18 3.55 3.89 5.19 8.04 Predictor Constant xi x2 x1x2 S 1.72225 R-sq=82.5% Analysis of Variance Source Regression Residual Coef 17.279 -6 . 368 -3 . 658 1.7067 SE Coef 7.167 2.260 1.364 0.4314 2.41 22.82 22.68 3.96 0.035 0.017 0.021 0.002 R-Sg (adj ) =77.8% DF MS 3 154.033 51.344 17.31 0.000 11 32.627 2.966 Error Total 14 186.660