First-Year College GPA Researchers at the College Board wanted to build a model that describes one's first-year college GPA. The researchers obtained the following model:

\[\hat{y}=0.06 x_{1}+0.07 x_{2}+0.18 x_{3}+0.29 x_{4}\]

where \(y\) represents the \(z\)-score for first-year college grade point average (GPA)

\(x_{1}\) represents the \(z\)-score on the math portion of the SAT

\(x_{2}\) represents the \(z\)-score on the critical reading portion of the SAT

\(x_{3}\) represents the \(z\)-score on the writing portion of the SAT

\(x_{4}\) represents the \(z\)-score of the student's high school grade point average (GPA)

(a) Suppose a student has a \(z\)-score of 1.52 on the math portion of the SAT. Explain what this result represents.

(b) What is the impact of a \(z\)-score of -1 for the student's high school GPA?

(c) Interpret the slope coefficient for high school GPA

(d) The coefficient of determination for this model is 0.24 . Interpret this value.

(e) The correlation coefficient between \(x_{2}\) and \(x_{3}\) is 0.71 . What might this suggest?

(f) Predict the \(z\)-score for first-year college GPA if \(x_{1}=-0.54, x_{2}=1.32, x_{3}=0.98\), and \(x_{4}=0.36\). Would we expect a student with these credentials to have an above or below average first-year college GPA?