Problem

Apply Chebyshev's Theorem to find the least possible fraction of the numbers in a data set lying within $\frac{6}{5}$ standard deviations of the mean. At least $\square$ of all numbers must lie within $\frac{6}{5}$ standard deviations from the mean. (Type an integer or a simplified fraction)

Solution

Step 1 :Chebyshev's theorem states that at least 1 - 1/k^2 of the data from a sample will fall within k standard deviations of the mean for all k > 1. In this case, k = \(\frac{6}{5}\).

Step 2 :We need to calculate 1 - 1/(\(\frac{6}{5}\))^2 to find the least possible fraction of the numbers in a data set lying within \(\frac{6}{5}\) standard deviations of the mean.

Step 3 :Final Answer: The least possible fraction of the numbers in a data set lying within \(\frac{6}{5}\) standard deviations of the mean is approximately 0.3056 or \(\boxed{\frac{61}{200}}\) when simplified as a fraction.

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Source: https://solvelyapp.com/problems/TC7ussU257/

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