An Economist Is Analysing the Incomes of Professionals (Physicians, Dentists $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$

Essay

An economist is analysing the incomes of professionals (physicians, dentists and lawyers). He realises that an important factor is the number of years of experience. However, he wants to know if there are differences among the three professional groups. He takes a random sample of 125 professionals and estimates the multiple regression model: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ .
where
y
= annual income (in $1000). $x _ { 1 }$ = years of experience. $x _ { 2 }$ = 1 if physician.
= 0 if not. $x _ { 3 }$ = 1 if dentist.
= 0 if not.
The computer output is shown below.
THE REGRESSION EQUATION IS $y =$ $71.65 + 2.07 x _ { 1 } + 10.16 x _ { 2 } - 7.44 x _ { 3 }$ . $$\begin{array} { | c | r r c | } \hline \text { Predictor } & \text { Coef } & \text { StDev } & T \\\hline \text { Constant } & 71.65 & 18.56 & 3.860 \\x _ { 1 } & 2.07 & 0.81 & 2.556 \\x _ { 2 } & 10.16 & 3.16 & 3.215 \\x _ { 3 } & - 7.44 & 2.85 & - 2.611 \\\hline\end{array}$$ S = 42.6 R-Sq = 30.9%. $$\begin{array}{l}\text { ANALYSIS OF VARIANCE }\\\begin{array} { | l | r c c c | } \hline \text { Source of Variation } & \text { df } & \text { SS } & \text { MS } & F \\\hline \text { Regression } & 3 & 98008 & 32669.333 & 18.008 \\\text { Error } & 121 & 219508 & 1814.116 & \\\hline \text { Total } & 124 & 317516 & & \\\hline\end{array}\end{array}$$ Is there enough evidence at the 5% significance level to conclude that income and experience are linearly related?