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.