Questions: The undergraduate grade point averages (UGPA) of students taking an admissions test in a recent year can be approximated by a normal distribution as shown in the figure. (a) What is the minimum UGPA that would still place a student in the top 10% of UGPAs? (b) Between what two values does the middle 50% of the UGPAs lie? (a) The minimum UGPA that would still place a student in the top 10% of UGPAs is (Round to two decimal places as needed.)

Solution Steps

To find the minimum UGPA that would place a student in the top 10%, we need to find the z-score that corresponds to the 90th percentile (since the top 10% corresponds to the area to the right of the z-score, which is equivalent to the area to the left being 90% or 0.90). Using a z-table or calculator, we find that the z-score corresponding to 0.90 is approximately 1.28.

We are given that the mean UGPA (μ) is 3.22 and the standard deviation (σ) is 0.19. We can use the z-score formula to find the corresponding UGPA (x):

z = (x - μ) / σ

Plugging in the values, we get:

1.28 = (x - 3.22) / 0.19

Multiply both sides by 0.19:

1.28 * 0.19 = x - 3.22

0.2432 = x - 3.22

Add 3.22 to both sides:

x = 3.22 + 0.2432

x = 3.4632

Rounding to two decimal places, we get 3.46.

Final Answer: The minimum UGPA that would still place a student in the top 10% of UGPAs is 3.46.