IF THE SMOOTHING FUNCTION IS GIVEN BY AN EQUATION hat (y)=alpha _(0)+alpha _(1)x , THEN THE FUNCTION DETERMINING THE COEFFICIENTS Select one: =sum _(i=1)^n(y_(i)-alpha _(1)x_(i))^2 =sum _(i=1)^n(alpha _(0)+alpha _(1)x_(i))^2 =sum _(i=1)^n(y_(i)-(alpha _(0)-alpha _(1)x_(i)))^2 =sum _(i=1)^n(y_(i)-alpha _(0))^2 =sum _(i=1)^n(y_(i)-(alpha _(0)+alpha _(1)x_(i)))^2

The correct answer is:<br />$=\sum _{i=1}^{n}(y_{i}-(\alpha _{0}+\alpha _{1}x_{i}))^{2}$<br /><br />Explanation:<br />The given equation is $\hat {y}=\alpha _{0}+\alpha _{1}x$, which represents a linear smoothing function. To determine the coefficients $\alpha_0$ and $\alpha_1$, we need to minimize the sum of the squared errors between the predicted values $\hat{y}$ and the actual values $y_i$.<br /><br />The sum of the squared errors is given by:<br />$\sum_{i=1}^{n} (y_i - \hat{y}_i)^2$<br /><br />Substituting the given equation, we get:<br />$\sum_{i=1}^{n} (y_i - (\alpha_0 + \alpha_1 x_i))^2$<br /><br />Therefore, the correct function for determining the coefficients is:<br />$=\sum_{i=1}^{n} (y_i - (\alpha_0 + \alpha_1 x_i))^2$