Chapter 8, Problem 40E

To determine

To find out what SAT score might be expected among freshmen who attained a first-semester GPA of $3.0$ .

Answer to Problem 40E

The SAT score will be $2029.52$ .

Explanation of Solution

In the question of exercise $38$ , shows the relationship between the GPA scores and the combined SAT scores. The scatterplot for the two variables showed the association to be reasonably linear. And it is given that:

  $\begin{array}{l}r=0.47\\ {\mu }_1=2.66\\ {\mu }_2=1833\\ {\sigma }_1=0.56\\ {\sigma }_2=123\end{array}$

Thus, the regression line will be:

  $\begin{array}{l}\text{<mover accent="true"><mo>G</mo><mo>^</mo></mover> PA}=\alpha +\beta (\text{SAT)}\\ =-1.262+0.0021(\text{SAT)}\end{array}$

Thus, the SAT score that might be expected among freshmen who attained a first-semester GPA of $3.0$ can be calculated as:

  $\begin{array}{l}\text{<mover accent="true"><mo>G</mo><mo>^</mo></mover> PA}=\alpha +\beta (\text{SAT)}\\ 3.0=-1.262+0.0021(\text{SAT)}\\ 4.262=0.0021(\text{SAT)}\\ \text{SAT}=2029.52\end{array}$