Unconditional Scatterplot
\(weeksal\) = Employee’s weekly salary (in dollars units).
\(hours\) = Number of hours the employee works in a week (in hours units).
\[weeksal = \beta_0 + \beta_1 hours + \varepsilon\]
This is simple linear regression.
\[weeksal = \beta_0 + \beta_1 hours + \varepsilon\]
\(\beta_1 = 0.45\) AND we failed to reject the null hypothesis that \(\beta_1=0\) \(\rightarrow\) We don’t have evidence here to say that we believe there is a true non-zero relationship between hours and weekly salary.
That doesn’t make sense?
This is an example of a confounding variable that is making it difficult to identify the “true” relationship between hours and weekly salary.
We want to understand the \(hours \rightarrow weeksal\) relationship independent of the counfounding role of gender.
\(female\) = An indicator for whether the employee identifies as female.
\[weeksal = \beta_0 + \beta_1 hours + \beta_2 female + \varepsilon\]
\(\beta_1 = 17.74\)
\(\beta_2 = -144.84\)
This is what multiple regression does: Allows you to look at relationships between variables independent of each other.
Controlling for gender, when there is a one hour increase in hours worked, there is an average $17.74 increase in weekly salary.
Controlling for hours, people who are female (as opposed to male) earn $144.84 less in weekly salary.
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