Consider the Model Where $x _ { 1 }$ Is a Quantitative Variable And

Multiple Choice

Consider the model
$$y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 1 } ^ { 2 } + \beta _ { 3 } x _ { 2 } + \beta _ { 4 } x _ { 3 } + \beta _ { 5 } x _ { 1 } x _ { 2 } + \beta _ { 6 } x _ { 1 } x _ { 3 } + \beta _ { 7 } x _ { 1 } 2 _ { x _ { 2 } } + \beta _ { 8 } x _ { 1 } x _ { 3 } + \varepsilon$$
where $x _ { 1 }$ is a quantitative variable and $x _ { 2 }$ and $x _ { 3 }$ are dummy variables describing a qualitative variable at three levels using the coding scheme
$$x _ { 2 } = \left\{ \begin{array} { l l } 1 & \text { if level } 2 \\0 & \text { otherwise }\end{array} \quad x _ { 3 } = \left\{ \begin{array} { l l } 1 & \text { if level } 3 \\0 & \text { otherwise }\end{array} \right. \right.$$
The resulting least squares prediction equation is
$$\hat { y } = 8.8 - 1.1 x _ { 1 } + 3.2 x _ { 1 } ^ { 2 } + 1.6 x _ { 2 } - 4.4 x _ { 3 } + .02 x _ { 1 } x _ { 2 } + 1.3 x _ { 1 } x _ { 3 } + .01 x _ { 1 } 2 ^ { 2 } - .06 x _ { 1 } x _ { 3 }$$
What is the equation of the response curve for $E ( y )$ when $x _ { 2 } = 0$ and $x _ { 3 } = 0$ ?

A) $\hat { y } = 8.8 - 1.1 x _ { 1 } + 3.2 x _ { 1 } { } ^ { 2 }$
B) $\hat { y } = 8.8 - 1.3 x _ { 1 } + 3.2 x _ { 1 } ^ { 2 }$
C) $\hat { y } = 8.8 - 1.6 x _ { 2 } - 4.4 x _ { 3 }$
D) $\hat { y } = 8.8 - .22 x _ { 1 } + 3.15 x _ { 1 } 2$