the undergraduate grade point averages (ugpa) of students taking an admissions test in a recent year can be…

Explanation:

Step1: Find the z - score for the top 15%

The area to the left of the z - score we want is (1 - 0.15=0.85). Looking up in the standard normal distribution table (z - table), the z - score (z) corresponding to an area of 0.85 is approximately (z = 1.04).

Step2: Use the z - score formula

The z - score formula is (z=\frac{x-\mu}{\sigma}), where (\mu = 3.34) is the mean and (\sigma=0.16) is the standard deviation. We want to solve for (x). Rearranging the formula gives (x=\mu + z\sigma). Substitute (\mu = 3.34), (z = 1.04) and (\sigma = 0.16) into the formula: (x=3.34+1.04\times0.16).

Step3: Calculate the value of (x)

First, calculate (1.04\times0.16 = 0.1664). Then (x=3.34 + 0.1664=3.5064\approx3.51).

For part (b):

Step1: Find the z - scores for the middle 50%

The middle 50% means that the area in the two tails is (1 - 0.50 = 0.50), so the area in each tail is (\frac{0.50}{2}=0.25). The z - score corresponding to an area of (0.25) in the left - hand tail is (z_1\approx - 0.67) and the z - score corresponding to an area of (0.25 + 0.50=0.75) (right - hand side of the middle 50%) is (z_2\approx0.67).

Step2: Use the z - score formula for the lower value

Using (z_1=\frac{x_1-\mu}{\sigma}), and solving for (x_1) gives (x_1=\mu+z_1\sigma). Substitute (\mu = 3.34), (z_1=-0.67) and (\sigma = 0.16): (x_1=3.34+( - 0.67)\times0.16=3.34 - 0.1072 = 3.2328\approx3.23).

Step3: Use the z - score formula for the upper value

Using (z_2=\frac{x_2-\mu}{\sigma}), and solving for (x_2) gives (x_2=\mu+z_2\sigma). Substitute (\mu = 3.34), (z_2 = 0.67) and (\sigma = 0.16): (x_2=3.34+0.67\times0.16=3.34 + 0.1072=3.4472\approx3.45).

Answer:

(a) 3.51 (b) 3.23 and 3.45