The article "Sensitivity Analysis of a 2.5 kW Proton Exchange Membrane Fuel Cell Stack by Statistical Method" (J. of Fuel Cell Sci. and Tech., 2009: 1-6) used regression analysis to investigate the relationship between fuel cell power (W) and the independent variables x1 = H2 pressure spsid, x2 = H2 flow (stoc), x3 = air pressure (psi) and x4 = airflow (stoc).
a. Here is Minitab output from fitting the model with the aforementioned independent variables as predictors (also fit by the authors of the cited article):

Does there appear to be a useful relationship between power and at least one of the predictors? Carry out a formal test of hypotheses.
b. Fitting the model with predictors x3, x4, and the interaction x3x4 gave R2 = .834. Does this model appear to be useful? Can an F test be used to compare this model to the model of (a)? Explain.
c. Fitting the model with predictors x1 - x4 as well as all second-order interactions gave R2 = .960 (this model was also fit by the investigators). Does it appear that at least one of the interaction predictors provides useful information about power over and above what is provided by the first-order predictors? State and test the appropriate hypotheses using a significance level of .05.

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Predictor Constant x1 x2 x3 Coef 1507.3 -4.282 7.46 -0.9162 90.60 SE Coef 206.8 4.969-0.86 0.407 62.11 7.29 0.000 0.12 0.907 0.6227-1.47 0.169 3.65 0.004 24.84 S -4.6885 R-Sq -59.6% R-Sq (adj ) -44.9% Source Regression Residual Error 11 27158 2469 Total DF MS 4 40048 10012 4.06 0.029 15 67206