A sample of 500 evening students revealed that their annual incomes were normally distributed with a mean income of $50,000 and a standard deviation of $4,000. What is the income that separates the top 25% from the lower 75% of the incomes?
A) $2,680
B) $47,320
C) $50,000
D) $52,680

Question

Quizplus · Accepted Answer

We need to find the income that separates the top 25% from the lower 75%, which corresponds to the z-score that separates the highest 25% from the lowest 75% in a standard normal distribution.

The z-score that corresponds to the 75th percentile is approximately 0.67, and the z-score that corresponds to the 25th percentile is approximately -0.67 (using a standard normal distribution table).

We can use the formula:

z = (x - μ) / σ

where z is the z-score, x is the income we want to find, μ is the mean income, and σ is the standard deviation.

Solving for x, we get:

x = z * σ + μ

x = 0.67 * 4,000 + 50,000

x = 52,680

Therefore, the income that separates the top 25% from the lower 75% of the incomes is $52,680. Choice D is the correct answer.

A Sample of 500 Evening Students Revealed That Their Annual