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.

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