The mean ft of a distribution is 20 and the standard deviati

The mean ft of a distribution is 20 and the standard deviation sigma is 2. Use Chebyshev\'s theorem to answer the following: At least what percentage of the values will fall between 10 and 30? At least what percentage of the values will fall between 12 and 28?

Solution

By Chebyshev\'s theorem, within k standard deviations from the mean lies 1-1/k^2 of the data.

a)

Let us test k for x = 30.

k = (x - u)/sigma = (30-20)/2 = 5

Thus,

1 - 1/k^2 = 1 - 1/5^2 = 0.96 or at least 96% [ANSWER]

b)

Let us test k for x = 28.

k = (x - u)/sigma = (28-20)/2 = 4

Thus,

1 - 1/k^2 = 1 - 1/4^2 = 0.9375 or at least 93.75% [ANSWER]