Professors at a local university earn an average salary of $80,000 with a standard deviation of $6,000. The salary distribution cannot be regarded as bell-shaped. What can be said about the percentage of salaries that are less than $68,000 or more than or more than $92,000?
A) It is at least 75%.
B) It is at most 75%.
C) It is at least 25%.
D) It is at most 25%.

Question

Quizplus · Accepted Answer

According to Chebyshev's theorem, for any distribution (regardless of its shape), at least $1 - \frac{1}{k^2}$ of the data values fall within k standard deviations from the mean. Here, $68,000 and $92,000 are 2 standard deviations away from the mean ($80,000 ± 2 \times $6,000). Thus, at least $1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4}$ or 75% of the salaries fall within this range. Therefore, at most 25% of the salaries are outside this range, making option D correct.

Professors at a Local University Earn an Average Salary of $80,000