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?

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.

Writing this problem in symbols instead of in words greatly simplifies it:

(2  2/3) * (1  1/5) * (1  1/2)

Write each quantity (inside each set of parentheses) as an improper fraction:

(8/3) * (6/5) * (3/2)

Now multiply the numerators thru:  8*6*3 / 3*5*2

Notice that we can reduce this by dividing the 8 by the 2 and dividing the 6 by the 3:                                                4*3*3             36
                                                         =  
                                                             5                  5