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Answer :
Final answer:
The z-score represents the number of standard deviations an individual's GPA falls above or below the mean GPA. The actual GPA corresponding to a particular z-score can be calculated given the mean and standard deviation of the GPAs. Actual minimum and maximum achievable GPA are set by educational institutions, but z-scores theoretically range from negative to positive infinity.
Explanation:
The z-score is a measure that indicates the number of standard deviations a data point, in this case, the undergraduate grade point averages (UGPA) of students taking an admissions test in a recent year, is from the mean of a distribution. The distribution in this case can be approximated by a normal distribution. To find the GPA corresponding to a particular z-score, we need to know the mean (μ) and standard deviation (σ) of the UGPA.
If 'z' represents the z-score, 'X' the actual GPA, 'μ' the mean GPA, and 'σ' the standard deviation of GPA, then the formula to calculate X, or the GPA corresponding to a specific z-score is X = zσ + μ.
(a) For a z-score of -2.5, the UGPA would be X = -2.5σ + μ (This would indicate a GPA that is 2.5 standard deviations below the mean).
(b) For a z-score of 1.8, the UGPA would be X = 1.8σ + μ (This would indicate a GPA that is 1.8 standard deviations above the mean).
Although we can calculate hypothetical UGPA values if we know the mean and standard deviation, the actual minimum and maximum GPA achievable are usually set by educational institutions and normally range from 0.0 to 4.0 (or 5.0 in some systems). Yet, in a normal distribution, the z-score would theoretically range from negative infinity to positive infinity.
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