Build a Regression Model
-Can Happiness Be Predicted? a Researcher

Build a Regression Model
-Can happiness be predicted? A researcher is investigating whether job satisfaction, income, and health could be good predictors of happiness. She selects a random sample of working adults and obtains the following information for each person. Each person is asked to rate their happiness and their job satisfaction on a scale of 1 to 10 . Each person is asked their annual income. Finally, each person's overall physical health is evaluated by a doctor and rated on a scale of 1 to 20 . The results are shown in the table.
$\begin{array}{c|c|c|c}\text { Happiness } & \text { Job satisfaction } & \begin{array}{c}\text { Annual Income } \\\text { (thousands of dollars) }\end{array} & \text { Health } \\\hline 8 & 6 & 38 & 14 \\6 & 5 & 47 & 17 \\4 & 2 & 19 & 12 \\7 & 5 & 22 & 11 \\2 & 3 & 36 & 5 \\9 & 7 & 44 & 14 \\4 & 3 & 75 & 8 \\6 & 8 & 24 & 13 \\7 & 4 & 52 & 14 \\3 & 3 & 27 & 5 \\5 & 4 & 105 & 12 \\8 & 9 & 44 & 7\end{array}$

(a) Construct the correlation matrix. Is there any reason to be concerned with collinearity?
(b) Find the least squares regression equation $\mathrm { y } = \mathrm { b } _ { 0 } + \mathrm { b } _ { 1 } \mathrm { x } _ { 1 } + \mathrm { b } _ { 2 } \mathrm { x } _ { 2 } + \mathrm { b } _ { 3 } \mathrm { x } _ { 3 }$ , where $\mathrm { x } _ { 1 }$ is job satisfaction, $\mathrm { x } _ { 2 }$ is income, $x _ { 3 }$ is health, and $y$ is the response variable "happiness".
(c) Test $\mathrm { H } _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = 0$ versus $\mathrm { H } _ { 1 }$ : at least one of the $\beta _ { \mathrm { i } } \neq 0$ at the $\alpha = 0.05$ level of significance.
(d) Test the hypotheses $\mathrm { H } _ { 0 } : \beta _ { 1 } = 0$ versus $\mathrm { H } _ { 1 } : \beta _ { 1 } \neq 0 , \mathrm { H } _ { 0 } : \beta _ { 2 } = 0$ versus $\mathrm { H } _ { 1 } : \beta _ { 2 } \neq 0$ , and $\mathrm { H } _ { 0 } : \beta _ { 3 } = 0$ versus $\mathrm { H } _ { 1 } : \beta _ { 3 } \neq 0$ at the $\alpha = 0.05$ level of significance.
Should any of the explanatory variables be removed from the model? If so, which one? Why?
(e) Determine the least squares regression equation with the explanatory variable identified in part (d) removed.
(f) Are both slope coefficients significantly different from zero? If not, remove the appropriate explanatory variable and compute the new least squares regression equation.
(g) How does the $\mathrm { P }$ -value for your final regression equation compare with the $\mathrm { P }$ -value for the original equation in part (b)? What does this imply?