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 is the proportion of faculty that earn more than $26,000 but less than $38,000?
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A. B. C. D. E.Explanation
26,000 and 38,000 are two standard deviations away from the mean. Using 1-(1/k^2) where k = 2, we get 0.75
To solve this question, we can use Chebyshev's Theorem, which provides a lower bound on the proportion of data that falls within a certain number of standard deviations from the mean in any distribution, regardless of its shape.
According to Chebyshev's Theorem, for any value "k" greater than 1, at least (1 - 1/k^2) proportion of data falls within "k" standard deviations from the mean. In this case, we need to determine the proportion of faculty members who earn more than $26,000 but less than $38,000, which represents a range of $12,000.
First, we calculate the number of standard deviations that $12,000 represents by dividing it by the standard deviation:
Number of standard deviations = $12,000 / $3,000 = 4
Now, we want to find the proportion of data that falls within 4 standard deviations from the mean. Using Chebyshev's Theorem, we know that at least (1 - 1/k^2) proportion of data falls within "k" standard deviations from the mean. In this case, k is 4, so:
Proportion within 4 standard deviations = 1 - 1/4^2 = 1 - 1/16 = 15/16
Therefore, at least 15/16 of the faculty members' incomes will fall within 4 standard deviations from the mean.
To determine the proportion of faculty members who earn more than $26,000 but less than $38,000, we need to calculate the range between these two values and divide it by the total range of 4 standard deviations:
Range between $26,000 and $38,000 = $38,000 - $26,000 = $12,000
Proportion of faculty members within the range = Range between $26,000 and $38,000 / Total range of 4 standard deviations = $12,000 / $12,000 = 1
Since the proportion of faculty members within the range is 1, it means that all faculty members will earn more than $26,000 but less than $38,000.
In summary, the proportion of faculty members who earn more than $26,000 but less than $38,000 is 100% or "At least 100%," which corresponds to answer choice D.