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 15% 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 15% of UGPAs is 3.47. (Round to two decimal places as needed.) (b) The middle 50% of UGPAs lies between on the low end and on the high end. (Round to two decimal places as needed.)

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 15% 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 15% of UGPAs is 3.47.
(Round to two decimal places as needed.)
(b) The middle 50% of UGPAs lies between on the low end and on the high end.
(Round to two decimal places as needed.)

Transcript text: 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 $15 \%$ 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 $15 \%$ of UGPAs is 3.47 . (Round to two decimal places as needed.) (b) The middle $50 \%$ of UGPAs lies between $\square$ on the low end and $\square$ on the high end. (Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Identify the given parameters

Mean (μ) = 3.28
Standard deviation (σ) = 0.18

Step 2: Determine the z-score for the top 15%

The top 15% corresponds to the 85th percentile.
Using a z-table or calculator, the z-score for the 85th percentile is approximately 1.04.

Step 3: Calculate the minimum UGPA for the top 15%

Use the z-score formula: $ X = μ + zσ $
$ X = 3.28 + (1.04 \times 0.18) $
$ X ≈ 3.28 + 0.1872 $
$ X ≈ 3.47 $

Final Answer

(a) The minimum UGPA that would still place a student in the top 15% of UGPAs is 3.47.

Step 4: Determine the z-scores for the middle 50%

The middle 50% corresponds to the 25th percentile to the 75th percentile.
Using a z-table or calculator, the z-scores for the 25th and 75th percentiles are approximately -0.67 and 0.67, respectively.

Step 5: Calculate the UGPAs for the middle 50%

For the 25th percentile: $ X = μ + zσ $
- $ X = 3.28 + (-0.67 \times 0.18) $
- $ X ≈ 3.28 - 0.1206 $
- $ X ≈ 3.16 $
For the 75th percentile: $ X = μ + zσ $
- $ X = 3.28 + (0.67 \times 0.18) $
- $ X ≈ 3.28 + 0.1206 $
- $ X ≈ 3.40 $

Final Answer

(b) The middle 50% of UGPAs lies between 3.16 on the low end and 3.40 on the high end.

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