Question

Question: GPAs of business students. Research scientists at the Educational Testing Service (ETS) used multiple regression analysis to model y, the final grade point average (GPA) of business and management doctoral students. A list of the potential independent variables measured for each doctoral student in the study follows:

Quantitative Graduate Management Aptitude Test (GMAT) score

Verbal GMAT score

Undergraduate GPA

First-year graduate GPA

Student cohort (i.e., year in which student entered doctoral program: year 1, year 3, or year 5)

1. Identify the variables as quantitative or qualitative.
2. For each quantitative variable, give your opinion on whether the variable is positively or negatively related to final GPA.
3. For each of the qualitative variables, set up the appropriate dummy variable.
4. Write a first-order, main effects model relating final GPA, y, to the five independent variables.
5. Interpret the b’s in the model, part d.
6. Write a first-order model for final GPA, y, that allows for a different slope for each student cohort.
7. For each quantitative independent variable in the model, part f, give the slope of the line (in terms of the b’s) for the year 1 cohort.

Short answer

Answer

1. 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.
2. 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.

1. 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, if student entered doctoral program in year 1; 0 otherwise,$x_5=1$, if student entered doctoral program in year 3; 0 otherwise$x_6=1$.
2. The first-order model relating the five variables to final GPA can be written as.$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_4+{\beta }_5{x}_5+{\beta }_6{x}_6$
3. Letrole="math" localid="1660741000929" ${\beta }_1$represents the changes in final GPA due to changes in Quantitative GMAT score. Let role="math" localid="1660741042705" ${\beta }_2$represents changes in the final GPA due to changes in verbal GMAT score. Let${\beta }_3$represents changes in the final GPA due to changes in undergraduate GPA. Let${\beta }_4$represents changes in the final GPA due to changes in first-year GPA. Let${\beta }_5$represents changes in the final grade when the students enter the doctoral program in year 1 andrepresents changes in the final GPA when the students enter doctoral program in year 3.
4. The first-order model equation with interaction can be written as role="math" localid="1660741334161" $$\begin{gathered} E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_4+{\beta }_5{x}_5+{\beta }_6{x}_6 \\ +{\beta }_7{x}_5{x}_1+{\beta }_8{x}_5{x}_2+{\beta }_9{x}_5{x}_3+{\beta }_{10}{x}_5{x}_4+{\beta }_{11}{x}_6{x}_1+{\beta }_{12}{x}_6{x}_2 \\ +{\beta }_{13}{x}_6{x}_3+{\beta }_{14}{x}_6{x}_4 \end{gathered}$$
5. For each quantitative independent variable in the model the slope for the year 1 cohort will be – for x₁, the slope coefficient will be $\left({\beta }_1+{\beta }_7\right)$, for x₂ , the slope coefficient will be$\left({\beta }_2+{\beta }_8\right)$ , for x₃, the slope coefficient will be $\left({\beta }_3+{\beta }_9\right)$, and for x₄ , the slope coefficient will be $\left({\beta }_4+{\beta }_{10}\right)$.

Step-by-step solution

Variable interpretation 

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

Variable interpretation

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.

Relation between dependent and quantitative variables 

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.

Relation between dependent and qualitative variables 

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

First-order model

d.

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

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

Where

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

Interpretation of βS 

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.

First-order model with interaction

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

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

Interpretation of  βS

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 $\left({\beta }_1+{\beta }_7\right)$, for x₂ , the slope coefficient will be ,$\left({\beta }_2+{\beta }_8\right)$for x₃ , the slope coefficient will be$\left({\beta }_3+{\beta }_9\right)$ and for x₄ , the slope coefficient will be$\left({\beta }_4+{\beta }_{10}\right)$ .