The Model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \beta _ { 4 } x _ { 4 }$

Essay

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \beta _ { 4 } x _ { 4 }$ was used to relate $E ( y )$ to a single qualitative variable, where
$x _ { 1 } = \left\{ \begin{array} { l l } 1 & \text { if level } 2 \\ 0 & \text { if not } \end{array} \quad x _ { 2 } = \left\{ \begin{array} { l l } 1 & \text { if level } 3 \\ 0 & \text { if not } \end{array} \right. \right.$

$x _ { 3 } = \left\{ \begin{array} { l l } 1 & \text { if level } 4 \\ 0 & \text { if not } \end{array} \quad x _ { 4 } = \left\{ \begin{array} { l l } 1 & \text { if level } 5 \\ 0 & \text { if not } \end{array} \right. \right.$
This model was fit to $n = 40$ data points and the following result was obtained:
$$\hat { y } = 14.5 + 3 x _ { 1 } - 4 x _ { 2 } + 10 x _ { 3 } + 8 x _ { 4 }$$
a. Use the least squares prediction equation to find the estimate of $E ( y )$ for each level of the qualitative variable.
b. Specify the null and alternative hypothesis you would use to test whether $E ( y )$ is the same for all levels of the independent variable.