Consider the model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ 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 $\hat { y } = 16.3 + 2.3 x _ { 1 } + 3.5 x _ { 2 } + 18 x _ { 3 }$. What is the response line (equation) for $E ( y )$ when $x _ { 2 } = 0$ and $x _ { 3 } = 1$ ?
A) $\hat { y } = 16.3 + 4.1 x _ { 1 }$
B) $\hat { y } = 18.6 + 2.3 x _ { 1 }$
C) $\hat { y } = 16.3 + 2.3 x _ { 1 }$
D) $\hat { y } = 18.1 + 2.3 x _ { 1 }$

Question

Quizplus · Accepted Answer

When $x_2 = 0$ and $x_3 = 1$, the equation simplifies to $\hat{y} = 16.3 + 2.3x_1 + 18x_3$. Substituting $x_3 = 1$ gives $\hat{y} = 16.3 + 2.3x_1 + 18$, which simplifies to $\hat{y} = 34.3 + 2.3x_1$, indicating a mistake in the provided options. However, based on the closest match principle and likely intended answer, D would be chosen if it were corrected to reflect the accurate calculation.

Consider the Model y=β0+β1x1+β2x2+β3x3+εy = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilony=β0​+β1​x1​+β2​x2​+β3​x3​+ε

Consider the Model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$