Question

A normal distribution has a mean of μ = 80 with σ = 20. What score separates the highest 15% of the distribution from the rest of the scores?

a) X = 95
b) X = 92
c) X = 100.8
d) X = 65

Answer 1

The score that separates the highest 15% is approximately 72.3.

None of the given options is correct

Step-by-step explanation:

To find the score that separates the highest 15% of the distribution from the rest, we can use the concept of z-scores and the standard normal distribution.

First, let's find the z-score corresponding to the highest 15% of the distribution. Since the normal distribution is symmetric, we know that 15% of the distribution lies above the mean (50% in total), leaving 35% below the mean.

Using a z-table or a calculator, we find that the z-score corresponding to 35% below the mean is approximately -0.385.

Next, we can use the z-score formula to find the corresponding score in the original distribution. The z-score formula is:

z = (X - μ) / σ

Rearranging the formula, we have:

X = μ + z * σ

Plugging in the values, we get:

X = 80 + (-0.385) * 20

X ≈ 80 - 7.7

X ≈ 72.3

Therefore, the score that separates the highest 15% of the distribution from the rest is approximately 72.3.

None of the answer choices provided match the correct score of approximately 72.3.