Chapter 15, Problem 44SE

The admissions officer for Clearwater College developed the following estimated regression equation relating the final college GPA to the student’s SAT mathematics score and highschool GPA.

ŷ = −1.41 + .0235x₁ + .00486x₂

where

x₁ = high-school grade point average

x₂ = SAT mathematics score

y = final college grade point average

1. a. Interpret the coefficients in this estimated regression equation.
2. b. Predict the final college GPA for a student who has a high-school average of 84 and a score of 540 on the SAT mathematics test.

To determine

Interpret the coefficients in the estimated regression equation.

Explanation of Solution

Calculation:

The estimated regression equation relating the final GPA to the student’s SAT mathematics score and high school is GPA is $\widehat{y}=-1.41+0.0235x_1+0.00486x_2$, where $x_1$ denotes the high school grade point average, $x_2$ denotes the SAT mathematics score and y denotes the final college grade point average.

Multiple linear regression model:

A multiple linear regression model is given as $\widehat{y}=b_0+b_1{x}_1+\mathrm{...}+b_p{x}_p$ where $\widehat{y}$ is the predicted value of response or dependent variable, and $x_1,x_2,\mathrm{...},x_p$ are the k predictor variables. The quantities $b_1,b_2,\mathrm{...},b_p$ are the estimated slopes corresponding to $x_1,x_2,\mathrm{...},x_p$ respectively and $b_0$ is the estimated intercept of the line, from the sample data.

Slope in a multiple regression equation:

The slope $b_i$ in a multiple regression equation is the amount of change in the response variable, $\widehat{y}$, due to unit increase in the corresponding predictor variable, $x_i$.

The ‘Coefficient’ column of the regression analysis output gives the slopes corresponding to the respective variables stored in the column ‘Variable’.

The coefficient or slope of $x_1$ in the regression model is $b_1=0.0235$.

The interpretation of the coefficient $b_1$ in the regression model is that the value of final college grade point average (y)increases by 0.0235unit for one unit increase in high school grade point average($x_1$), provided the effect of SAT mathematical score ($x_2$ ) is constant.

The coefficient or slope of $x_2$ in the regression model is $b_2=0.00486$.

The interpretation of the coefficient $b_2$ in the regression model is that the value of final college grade point average (y)increases by 0.00486unit for one unit increase in SAT mathematical score ($x_2$ ), provided the effect of high school grade point average($x_1$) is constant.

To determine

Predict final college GPA for a student who has a high-school average of 84 and a score of 540 on the SAT mathematical test.

Answer to Problem 44SE

The predicted final college GPA for a student who has a high-school average of 84 and a score of 540 on the SAT mathematical test is 3.19.

Explanation of Solution

Calculation:

The high school average of 84 implies that, $x_1=84$ and SAT mathematical score 540 implies that $x_2=540$.

The regression equation is $\widehat{y}=-1.41+0.0235x_1+0.00486x_2$.

For $x_1=84$ and $x_2=540$, predicted value of y is,

$\begin{array}{c}\widehat{y}=-1.41+0.0235\left(84\right)+0.00486\left(540\right)\\ =-1.41+1.974+2.6244\\ \approx 3.19\end{array}$

Thus, the predictedfinal college GPA for a student who has a high-school average of 84 and a score of 540 on the SAT mathematical test is 3.19.