The following model was used to relate E(y) to a single qualitative variable with four levels:$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}$where

${\mathbf{x}}_{\mathbf{1}}\mathbf{=}\left(\begin{array}{l}\mathbf{1}\mathbf{if}\mathbf{level}\mathbf{2}\\ \mathbf{0}\mathbf{if}\mathbf{not}\end{array}\right){\mathbf{x}}_{\mathbf{2}}\mathbf{=}\left(\begin{array}{l}\mathbf{1}\mathbf{if}\mathbf{level}\mathbf{3}\\ \mathbf{0}\mathbf{if}\mathbf{not}\end{array}\right){\mathbf{x}}_{\mathbf{3}}\mathbf{=}\left(\begin{array}{l}\mathbf{1}\mathbf{if}\mathbf{level}\mathbf{4}\\ \mathbf{0}\mathbf{if}\mathbf{not}\end{array}\right)$

This model was fit to n = 30 data points, and the following result was obtained:

$\hat{\mathbf{y}}\mathbf{=}\mathbf{10}\mathbf{.}\mathbf{2}\mathbf{-}\mathbf{4}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}\mathbf{12}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathbf{x}}_{\mathbf{3}}$

1. Use the least-squares prediction equation to find the estimate of E(y) for each level of the qualitative independent variable.
2. Specify the null and alternative hypotheses you would use to test whether E(y) is the same for all four levels of the independent variable.

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