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.)
failed

Solution

failed
failed

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.

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Combine. 6 sqrt(80)-sqrt(5) 6 sqrt(80)-sqrt(5)= (Simplify your answer. Type an exact answer, using radicals as needed.)
failedAnswered
Which of the following is an appropriate tip from ChooseMyPlate.gov to help parents and caregivers make healthy choices for children?
failedAnswered
Which of the following is a characteristic of socialism? (A) Formal economic policies exist (B) A market economy and a command economy exists (C) High taxes exist (D) Government owns key industries (E) All the above.
failedAnswered
Simplify: 8 √128
failedAnswered
Given the function f and point a below, complete parts (a)-(c). f(x)=5 x^2, x ≥ 0, a=2 f'(x)=√5 b. Graph f and f^(-1) together. Choose the correct graph below. A. B. C. D.