Question

The following statement appeared in the article "Dimensions of Adjustment Among College Women" (Journal of College Student Development \([1998]: 364):\) Regression analyses indicated that academic adjustment and race made independent contributions to academic achievement, as measured by current GPA. Suppose $$ \begin{aligned} y &=\text { current GPA } \\ x_{1} &=\text { academic adjustment score } \\ x_{2} &=\text { race (with white }=0 \text { , other }=1) \end{aligned} $$ What multiple regression model is suggested by the statement? Did you include an interaction term in the model? Why or why not?

Short answer

The multiple regression model is \(y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon\). There's no need to include an interaction term as the problem states the variables contribute independently to the dependent variable.

Step-by-step solution

Define the variables

Define the given variables in the context of the problem: \(y\) for current GPA, \(x_1\) for academic adjustment score, and \(x_2\) for race where white is represented as 0, and others as 1.

Formulate the Regression Model

Form the multiple regression model based on the problem statement using the formula \(y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon\), where \(y\) is the dependent variable (current GPA), \(\beta_0\) is the y-intercept, \(\beta_1\) and \(\beta_2\) are the coefficients of the independent variables \(x_1\) (academic adjustment score) and \(x_2\) (race), respectively, and \(\epsilon\) represents the error term. Each of the independent variables \(x_1\) and \(x_2\) are contributing independently as mentioned in the problem statement.

Decide on the Interaction Term

Determine whether to include an interaction term (\(\beta_3x_1x_2\)) in the model. An interaction term would indicate that the effect of one predictor variable depends on the value of another predictor variable. However, the problem statement says that the variables contribute independently to the GPA score, so the interaction term should not be included.