Final answer:
To determine the percentage of students with a GPA higher than 3.8, we calculate a z-score and use the standard normal distribution table. The calculated z-score of 2.33 indicates that approximately 1% of students have a GPA higher than 3.8, which is not reflected in the given options.
Step-by-step explanation:
Determining the Percentage of Students with a High GPA
To calculate the percentage of students with a GPA higher than 3.8, we use z-score analysis, which measures how many standard deviations an observation is away from the mean. Given a mean (μ) of 3.1 and a standard deviation (σ) of 0.3, we first calculate the z-score for a GPA of 3.8 using the formula z = (x - μ) / σ, where x is the GPA in question.
For a GPA of 3.8, the z-score is z = (3.8 - 3.1) / 0.3 = 2.33. Next, we refer to a standard normal distribution table to find the area under the curve to the right of this z-score. This area represents the percentage of students with a GPA higher than 3.8. Z-score tables typically provide the area to the left, so we subtract this value from 1 to get the area to the right. For a z-score of 2.33, the table gives us an area to the left of approximately 0.99, implying that roughly 1% of students have a GPA higher than 3.8. Therefore, none of the given options (a 5%, b 10%, c 15%, d 20%) are correct.