?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

Chapter 14, Problem 5

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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)

Source: Kobrin, J., Sinharay, S., Haberman, S.J., & Chajewski, M. An Investigation of the Fit of Linear Regression Models to Data from an SAT Validity Study. College Board Research Report 2011–13.

(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?

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