Question

A sample of assistant professors on the business faculty at state supported institutions in Ohio revealed the mean income to be $32,000 for 9 months with a standard deviation of $3,000. Using Chebyshev's Theorem, what proportion of the faculty earns more than $26,000 but less than $38,000?

a) At least 50%
b) At least 25%
c) At least 75%
d) At least 100%

Answer 1

Using Chebyshev's Theorem, at least 75% of the faculty members earn between $26,000 and $38,000, which is option (c).

Step-by-step explanation:

The student's question is asking to apply Chebyshev's Theorem to determine the proportion of faculty members who earn more than $26,000 but less than $38,000 based on the provided mean income and standard deviation.

Chebyshev's Theorem states that for any number k greater than 1, at least (1 - 1/k^2) of the data values will fall within k standard deviations of the mean. In this case, $26,000 and $38,000 are each one standard deviation away from the mean ($32,000 ± $3,000).Using Chebyshev's Theorem, at least 75% of the faculty members earn between $26,000 and $38,000, which is option (c).

Therefore, k is 2 (since $32,000 ± 2 * $3,000 encompasses the range $26,000 to $38,000). Applying Chebyshev's Theorem, at least 1 - 1/2^2, which is 1 - 1/4, or at least 75% of the faculty salaries fall within this range. Therefore, the correct option is (c) At least 75%.