Question

Question: Consider the model:

$\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}\mathbf{\epsilon }$

where x₁ is a quantitative variable and x₂ and x₃ are dummy variables describing a qualitative variable at three levels using the coding scheme

role="math" localid="1649846492724" $$\begin{gathered} {\mathbf{x}}_{\mathbf{2}}\mathbf{=}\left(\mathbf{1} \mathbf{if}\mathbf{level} \mathbf{2} \\ \mathbf{0} \mathbf{otherwise}\right) {\mathbf{x}}_{\mathbf{3}}\mathbf{=}\left(\mathbf{1} \mathbf{if} \mathbf{level} \mathbf{3} \\ \mathbf{0} \mathbf{otherwise}\right) \end{gathered}$$

The resulting least squares prediction equation is $\hat{\mathbf{y}}\mathbf{=}\mathbf{44}\mathbf{.}\mathbf{8}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{2}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}\mathbf{9}\mathbf{.}\mathbf{4}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{15}\mathbf{.}\mathbf{6}{\mathbf{x}}_{\mathbf{3}}$

a. What is the response line (equation) for E(y) when x₂ = x₃ = 0? When x₂ = 1 and x₃ = 0? When x₂ = 0 and x₃ = 1?

b. What is the least squares prediction equation associated with level 1? Level 2? Level 3? Plot these on the same graph.

Short answer

a. The response lines for when x₂ = x₃ = 0 is $\hat{\mathrm{y}}=44.8+2.2x_1$. The response line when x₂ = 1 and x₃ = 0 is $\hat{y}=54.2+2.2x_1$. The response line for when x₂ = 0 and x₃ = 1 is $\hat{y}=60.4+2.2x_1$.

b. Graph

Step-by-step solution

Response lines

The response line for when x₂ = x₃ = 0 will be

$$\begin{gathered} \hat{y}=44.8+2.2x_1+9.4\left(0\right)+15.6\left(0\right) \\ \hat{y}=44.8+2.2x_1 \end{gathered}$$

The response line for when x₂ = 1 and x₃ = 0 will be

$$\begin{gathered} \hat{y}=44.8+2.2x_1+9.4\left(1\right)+15.6\left(0\right) \\ \hat{y}=(44.8+9.4)+2.2x_1 \\ \hat{y}=54.2+2.2x_1 \end{gathered}$$

The response line for when x₂ = 0 and x₃ = 1 will be

$$\begin{gathered} \hat{y}=44.8+2.2x_1+9.4\left(0\right)+15.6\left(1\right) \\ \hat{y}=(44.8+15.6)+2.2x_1 \\ \hat{y}=60.4+2.2x_1 \end{gathered}$$

Graph

The response line for when x₂ = x₃ = 0 will be

$$\begin{gathered} \hat{y}=44.8+2.2x_1+9.4\left(0\right)+15.6\left(0\right) \\ \hat{y}=44.8+2.2x_1 \end{gathered}$$

The response line for when x₂ = 1 and x₃ = 0 will be

role="math" localid="1649847635365" $$\begin{gathered} \hat{y}=44.8+2.2x_1+9.4\left(1\right)+15.6\left(0\right) \\ \hat{y}=(44.8+9.4)+2.2x_1 \\ \hat{y}=54.2+2.2x_1 \end{gathered}$$

The response line for when x₂ = 0 and x₃ = 1 will be

$$\begin{gathered} \hat{y}=44.8+2.2x_1+9.4\left(0\right)+15.6\left(1\right) \\ \hat{y}=(44.8+15.6)+2.2x_1 \\ \hat{y}=60.4+2.2x_1 \end{gathered}$$