Consider the polynomial regression model of degree Y_i = ?₀ + ?₁X_i+ ?₂ $x _ { i } ^ { 2 }$ + ...+ ?_r $x _ { i } ^ { r }$ + u_i. According to the null hypothesis that the regression is linear and the alternative that is a polynomial of degree r corresponds to A)H₀: ?_r = 0 vs. ?_r ? 0 B)H₀: ?_r = 0 vs. ?₁ ? 0 C)H₀: ?₃ = 0, ..., ?_r = 0, vs. H₁: all ?_j ? 0, j = 3, ..., r D)H₀: ?₂ = 0, ?₃ = 0 ..., ?_r = 0, vs. H₁: at least one ?_j ? 0, j = 2, ..., r

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The correct hypothesis for testing if a regression model is linear (degree 1) against an alternative that it is a polynomial of degree r involves testing if all coefficients of terms higher than the first degree (x) are zero under the null hypothesis. This means we are testing if the model reduces to a linear model (only the first-degree term is non-zero) versus the alternative that at least one higher-degree term has a non-zero coefficient, indicating a polynomial relationship.

Consider the Polynomial Regression Model of Degree Yi = ?0 xi2x _ { i } ^ { 2 }xi2​

Consider the Polynomial Regression Model of Degree Y_i = ?₀ $x _ { i } ^ { 2 }$