91Ó°ÊÓ

There are five independent variable provided in the problem. A dependent variable the final grade point average is dependent on independent variables.

a.

Variables in the model can be identified as quantitative or qualitative.

The independent variables quantitative GMAT score, verbal GMAT score, undergraduate GPA, and first-year graduate GPA are quantitative variables in the model while the student cohort is a qualitative variable with 3 levels as year 1, year 3, and year 5.

b.

The final grade point average (GPA) of business and management doctoral students (y) is related to the independent variables in following manner:

• Quantitative GMAT score is positively related.
• Verbal GMAT score is positively related.
• Undergraduate GPA is positively related.
• First-year graduate GPA is positively related.

c.

The qualitative here is student cohort with 3 levels of year 1, year 3, and year 5. The variables to be introduced in the model to represent the 3 levels will be

Therefore, say $x_5=1$, if student entered doctoral program in year 1; 0 otherwise

, if $x_6=1$student entered doctoral program in year 3; 0 otherwise

d.

The first-order model relating the five variables to final GPA can be written as

$y={\mathrm{β}}_0+{\mathrm{β}}_1{x}_1+{\mathrm{β}}_2{x}_2+{\mathrm{β}}_3{x}_3+{\mathrm{β}}_4{x}_4+{\mathrm{β}}_5{x}_5+{\mathrm{β}}_6{x}_6$

Where

$$\begin{gathered} x_1=\text{quantitative GMAT score} \\ {x}_2=\text{verbal GMAT score} \\ {x}_3=under\text{graduate GPA} \\ {x}_4=\text{first - year graduate GPA} \\ {x}_5=1\text{,if student entered doctoral program in year}1 \text{;}0\text{otherwise} \\ {x}_6=1\text{, if student entered doctoral program in year}3 \text{;} 0\text{otherwise} \end{gathered}$$

e.

A slope coefficient represents the changes in final GPA due to changes in Quantitative GMAT score. A slope coefficient represents changes in the final GPA due to changes in verbal GMAT score. A slope coefficient represents changes in the final GPA due to changes in undergraduate GPA. A slope coefficient represents changes in the final GPA due to changes in first-year GPA. A slope coefficient represents changes in the final grade when the students enter the doctoral program in year 1 and a slope coefficient represents changes in the final GPA when the students enter doctoral program in year 3.

f.

A first-order model equation for the final GPA, y, that allows for a different slope for each student cohort is a model with interaction terms included in the model.

Mathematically, the equation can be written as

$$\begin{gathered} E\left(y\right)={\mathrm{β}}_0+{\mathrm{β}}_1{x}_1+{\mathrm{β}}_2{x}_2+{\mathrm{β}}_3{x}_3+{\mathrm{β}}_4{x}_4+{\mathrm{β}}_5{x}_5+{\mathrm{β}}_6{x}_6 \\ +{\mathrm{β}}_7{x}_5{x}_1+{\mathrm{β}}_8{x}_5{x}_2+{\mathrm{β}}_9{x}_5{x}_3+{\mathrm{β}}_{10}{x}_5{x}_4+{\mathrm{β}}_{11}{x}_6{x}_1+{\mathrm{β}}_{12}{x}_6{x}_2 \\ +{\mathrm{β}}_{13}{x}_6{x}_3+{\mathrm{β}}_{14}{x}_6{x}_4 \end{gathered}$$

g.

For each quantitative independent variable in the model the slope for the year 1 cohort will be –for ,x₁ the slope coefficient will be $\left({\mathrm{β}}_1+{\mathrm{β}}_7\right)$, for x₂ , the slope coefficient will be ,$\left({\mathrm{β}}_2+{\mathrm{β}}_8\right)$for x₃ , the slope coefficient will be$\left({\mathrm{β}}_3+{\mathrm{β}}_9\right)$ and for x₄ , the slope coefficient will be$\left({\mathrm{β}}_4+{\mathrm{β}}_{10}\right)$ .