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
$$\begin{array} { l l } x _ { 1 } = \left\{ \begin{array} { l l } 1 & \text { if level 2 } \\0 & \text { if not }\end{array} \right. & x _ { 2 } = \left\{ \begin{array} { l l } 1 & \text { if level } 3 \\0 & \text { if not }\end{array} \right. \\x _ { 3 } = \left\{ \begin{array} { l l l } 1 & \text { if level } 4 & x _ { 4 } \\0 & \text { if not }\end{array} \right. & \left\{ \begin{array} { l l } 1 & \text { if level } 5 \\0 & \text { if not }\end{array} \right.\end{array}$$
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. 3 Test if Model is Useful for Predicting y