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

Answer:

d. It is at most 25 percent.

Step-by-step 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:

$U=\frac{\ 92,000-\ 80,000}{\ 6,000}=2\\L = \frac{\ 80,000-\ 68,000}{\ 6,000}= 2$

For k = 2, applying Chebyshev's inequality:

$P( \ 68,000 \leq X \leq \ 92,000)= \frac{1}{2^2} = 0.25$

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