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?

Question

Mathematics

Carl Wagner

23 October, 10:28

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?

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Julius · Answer 1

75% of the faculty earns more than $66,000, but less than $78,000.

Step-by-step explanation:

Chebyshev's Theorem states that:

75% of the measures lie within 2 standard deviations of the mean

89% of the measures lie within 3 standard deviations of the mean

In this problem, we have that:

Mean = 72,000

Standard deviation = 3,000

Using Chebyshev's theorem, what proportion of the faculty earns more than $66,000, but less than $78,000?

66,000 = 72,000 - 2*3,000

So 66,000 is two standard deviations below the mean

78,000 = 72,000 + 2*3,000

So 78,000 is two standard deviations above the mean

So, by Chebyshev's Theorem, 75% of the faculty earns more than $66,000, but less than $78,000.