The Lowest Score Required for a Responsible Position

D

Find the z value representing 44% of the area under the curve. From the z tables, z = 1.55. Using z = (x-u)/sigma. 1.55 = (x-500)/50. x = 577.5

To determine the lowest score a college graduate must earn to qualify for a responsible position, we need to find the score that corresponds to the upper 6 percent of the distribution.

Given that the distribution of test scores is normal, we can use the properties of the standard normal distribution (also known as the Z-distribution) to solve this problem. The standard normal distribution has a mean of 0 and a standard deviation of 1.

First, we need to find the Z-score that corresponds to the upper 6 percent of the distribution. The upper tail of the distribution represents the percentage above a certain Z-score. We can use a Z-table or a statistical calculator to find this value.

The upper 6 percent of the distribution corresponds to a Z-score that represents the area to the left of it equal to 94 percent (100 percent - 6 percent). Consulting a Z-table, we can find that the Z-score for a cumulative area of 0.94 is approximately 1.555.

Now, we can use the Z-score formula to find the raw score (test score) corresponding to this Z-score. The Z-score formula is:

Z = (X - μ) / σ

Where: Z is the Z-score, X is the raw score (test score), μ is the population mean, and σ is the population standard deviation.

We are given the mean test score of 500 and the standard deviation of 50. Plugging in these values and the Z-score of 1.555, we can solve for X:

1.555 = (X - 500) / 50

Simplifying the equation:

1.555 * 50 = X - 500 77.75 = X - 500 X = 77.75 + 500 X = 577.75

Rounding up to the nearest whole number, the lowest score a college graduate must earn to qualify for a responsible position is 578.

Therefore, the correct answer is option D: 578.