Solution Found!
The following model was used to relate E(y) to a
Chapter 12, Problem 69E(choose chapter or problem)
The following model was used to relate E(y) to a single quantitative variable with four levels:
\(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}\)
where
\(x_{1}=\left\{\begin{array}{ll} 1 & \text { if level } 2 \\ 0 & \text { if not } \end{array} x_{2}=\left\{\begin{array}{l} 1 \text { if level } 3 \\ 0 \text { if not } \end{array} \quad x_{3}=\left\{\begin{array}{l} 1 \text { if level } 4 \\ 0 \text { if not } \end{array}\right.\right.\right.\)
This model was fit to n = 30 data points, and the following result was obtained:
\(\hat{y}=10.2-4 x_{1}+12 x_{2}+2 x_{3}\)
a. Use the least squares prediction equation to find the estimate of E(y) for each level of the qualitative independent variable.
b. 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.
Questions & Answers
QUESTION:
The following model was used to relate E(y) to a single quantitative variable with four levels:
\(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}\)
where
\(x_{1}=\left\{\begin{array}{ll} 1 & \text { if level } 2 \\ 0 & \text { if not } \end{array} x_{2}=\left\{\begin{array}{l} 1 \text { if level } 3 \\ 0 \text { if not } \end{array} \quad x_{3}=\left\{\begin{array}{l} 1 \text { if level } 4 \\ 0 \text { if not } \end{array}\right.\right.\right.\)
This model was fit to n = 30 data points, and the following result was obtained:
\(\hat{y}=10.2-4 x_{1}+12 x_{2}+2 x_{3}\)
a. Use the least squares prediction equation to find the estimate of E(y) for each level of the qualitative independent variable.
b. 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.
ANSWER:Step 1 of 4
For a single qualitative variable, we have the model 
Where,
For the above model, we have the fitted model as
Sample number of points n = 30