An economist was interested in modeling the relation among annual income,
level of education, and work experience. The level of education is the number
of years of education beyond eighth grade, so 1 represents completing I year
of high school, 8 means completing 4 years of college, and so on. Work
experience is the number of years employed in the current profession. From a
random sample of 12 individuals, he obtained the following data:
$$ \begin{array}{ccc} \begin{array}{l} \text { Work Experience } \\ \text {
(years) } \end{array} & \begin{array}{l} \text { Level of } \\ \text {
Education } \end{array} & \begin{array}{l} \text { Annual Income } \\ \text {
(\$ thousands) } \end{array} \\
\hline 21 & 6 & 34.7 \\ \hline 14 & 3 & 17.9 \\ \hline 4 & 8 & 22.7 \\\
\hline 16 & 8 & 63.1 \\ \hline 12 & 4 & 33.0 \\ \hline 20 & 4 & 41.4 \\\
\hline 25 & 1 & 20.7 \\ \hline 8 & 3 & 14.6 \\ \hline 24 & 12 & 97.3 \\\
\hline 28 & 9 & 72.1 \\ \hline 4 & 11 & 49.1 \\ \hline 15 & 4 & 52.0 \\\
\hline \end{array} $$
(a) Construct a correlation matrix between work experience, level of
education, and annual income. Is there any reason to be concerned with
multicollinearity based on the correlation matrix?
(b) Find the least-squares regression equation \(\hat{y}=b_{0}+\) \(b_{1}
x_{1}+b_{2} x_{2},\) where \(x_{1}\) is work experience, \(x_{2}\) is level of
education, and \(y\) is the response variable, annual income.
(c) Draw residual plots and a boxplot of the residuals to assess the adequacy
of the model.
(d) Interpret the regression coefficients for the least-squares regression
equation.
(e) Determine and interpret \(R^{2}\) and the adjusted \(R^{2}\)
(f) Test \(H_{0}: \beta_{1}=\beta_{2}=0\) versus \(H_{1} ;\) at least one of the
\(\beta_{i} \neq 0\) at the \(\alpha=0.05\) level of significance.
(g) Test the hypotheses \(H_{0}: \beta_{1}=0\) versus \(H_{1}: \beta_{1} \neq 0\)
and \(H_{0}: \beta_{2}=0\) versus \(H_{1}: \beta_{2} \neq 0\) at the \(\alpha=0.05\)
level of significance.
(h) Predict the mean income of all individuals whose experience is 12 years
and level of education is 4
(i) Predict the income of a single individual whose experience is 12 years and
level of education is 4
(j) Construct \(95 \%\) confidence and prediction intervals for income when
experience is 12 years and level of education is 4
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