According to Chebyshev’s theorem, what proportion of a distribution will be with k=4 standard deviations of the mean? Show all work as to how to find this
To find the proportion of a distribution within k standard deviations of the mean using Chebyshev's theorem, you can follow these steps:
Step 1: Determine the value of k which represents the number of standard deviations from the mean.
Step 2: Apply Chebyshev's theorem formula, which states that at least (1 - 1/k^2) of the data lies within k standard deviations of the mean. In this case, k=4.
Step 3: Calculate the proportion by subtracting the complement of the proportion within k standard deviations from 1. The complement is (1 - (1 - 1/k^2)).
Let's calculate it step by step:
Step 1: Given the value of k = 4, we know that we need to determine the proportion of the distribution within 4 standard deviations of the mean.
Step 2: Applying the Chebyshev's theorem formula, we have:
Proportion within k standard deviations = 1 - 1/k^2
Proportion within 4 standard deviations = 1 - 1/4^2
= 1 - 1/16
= 15/16
Step 3: Calculating the complement and finding the proportion:
Complement = 1 - (1 - 1/k^2)
Complement = 1 - (1 - 1/4^2)
Complement = 1 - (1 - 1/16)
Complement = 1 - (15/16)
Complement = 1/16
Proportion = 1 - Complement = 1 - (1/16) = 15/16
Therefore, according to Chebyshev's theorem, approximately 15/16 (or 93.75%) of the distribution will be within 4 standard deviations of the mean.
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