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

According to Chebyshev's theorem, for any value k greater than 1, at least (1-1/k^2) of the data values will fall within k standard deviations of the mean.

Let's use k=2, because we know that at least 75% of the data values will fall within 2 standard deviations of the mean.

The range of interest is from $66,000 to $78,000, which is 2 standard deviations away from the mean of $72,000.

So, at least (1-1/2^2) = 75% of the assistant professors earn between $66,000 and $78,000. Therefore, the answer is C.

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