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 }^{ 2} { 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 } x _ { 2 } - .06 x _ { 1 }^{2} 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 }

\hat { y } = 8.8 - 1.3 x _ { 1 } + 3.2 x _ { 1 } { } ^ { 2 }

y = 8.8 - .22 x _ { 1 } + 3.15 x _ { 1 } { } ^ { 2 }

y = 8.8 - 1.6 x _ { 2 } - 4.4 x _ { 3 }

The Model E(y)=β0+β1x1+β2x2+β3x3E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 }E(y)=β0​+β1​x1​+β2​x2​+β3​x3​

Essay