Question

The annual salaries for people in a particular profession are known to be normally distributed with a mean of $54,800 and a standard deviation of $2,600.

What percentage of people in this profession earn annual salaries between $54,000 and $57,000? Round your result to one decimal place.

Answer 1

Approximately 4.5% of people in this profession earn annual salaries between $54,000 and $57,000.

Step-by-step explanation:

To find the percentage of people in this profession who earn annual salaries between $54,000 and $57,000, we need to calculate the z-scores for these two salary values and then use the standard normal distribution table.

First, we calculate the z-score for $54,000:

z = (X - μ) / σ

z = (54000 - 54800) / 2600

z = -0.030769

Next, we calculate the z-score for $57,000:

z = (X - μ) / σ

z = (57000 - 54800) / 2600

z = 0.084615

Now, we look up the corresponding cumulative probabilities for these z-scores in the standard normal distribution table.

The probability for a z-score of -0.030769 is 0.4875, and the probability for a z-score of 0.084615 is 0.5326.

To find the percentage between these two salaries, we subtract the probability for $54,000 from the probability for $57,000:

0.5326 - 0.4875 = 0.0451

Multiplying this result by 100, we find that approximately 4.5% of people in this profession earn annual salaries between $54,000 and $57,000.