Question

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 $92,000?

a. It is at least 75 percent.
b. It is at least 55 percent.
c. It is at least 25 percent.
d. It is at most 25 percent.

Answer 1

d. It is at most 25 percent.

Explanation:

According to Chebyshev's inequality, for a given number of standard deviations, k, no more than 1/k² can be more than k standard deviations from the mean. In this situation the amount of standard deviations from the mean of the upper and lower bound of salaries are:

`U=(\$92,000-\$80,000)/(\$6,000)=2\\L = (\$80,000-\$68,000)/(\$6,000)= 2`

For k = 2, applying Chebyshev's inequality:

`P( \$68,000 \leq X \leq \$92,000)= (1)/(2^2) = 0.25`

Therefore, at most 25% of the salaries are less than $68,000 or more than $92,000.