Question: Using \(z\) -scores for grades. At one university, the students are given \(z\) -scores at the end of each semester rather than the traditional GPAs. The mean and standard deviation of all students' cumulative GPAs, on which the \(z\) -scores are based, are 2.7 and .5, respectively.

1. Translate each of the following \(z\) -scores to corresponding GPA: \(z=2.0, z=-1.0, z=.5, z=-2.5\).
2. Students with \(z\) -scores below \(-1.6\) are put on probation. What is the corresponding probationary GPA?
3. The president of the university wishes to graduate the top \(16 \%\) of the students with cum laude honors and the top \(2.5 \%\) with summa cum laude honors. Where (approximately) should the limits be set in terms of \(z\) -scores? In terms of GPAs? What assumption, if any, did you make about the distribution of the GPAs at the university?

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