A sample of assistant professors on the business faculty at state-supported institutions in Ohio revealed the mean income to be $72,000 for nine months, with a standard deviation of $3,000. Using Chebyshev's Theorem, what proportion of the faculty earns more than $66,000, but less than $78,000?
A)At least 50%
B)At least 25%
C)At least 75%
D)At least 100%

Question

Quizplus · Accepted Answer

Chebyshev's Theorem states that for any dataset, regardless of distribution, at least $1 - \frac{1}{k^2}$ of the data values will fall within $k$ standard deviations from the mean. Here, $72,000 is the mean, and $3,000 is the standard deviation. The range from $66,000 to $78,000 is $\pm2$ standard deviations from the mean ($72,000 \pm 2 \times 3,000$). Thus, $k = 2$, and applying Chebyshev's Theorem gives at least $1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4}$ or 75% of the faculty salaries fall within this range.

A Sample of Assistant Professors on the Business Faculty at State-Supported