Chebyshev's Theorem guarantee that we will find at least 89% of smoking males in the life expectancy range [52.6, 84.4].
In probability theory, Chebyshev’s inequality guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean.
Let X be a random variable with mean μ with a finite variance σ², then for any real number k > 0, P(|X-μ| < kσ) ≥ 1-1/k².
Given,
Mean life expectancy = μ = 68.5 years
Standard deviation = σ = 5.3 years
P(|X-μ| < kσ) ≥ 1-1/k²
P(|X-68.5| < 5.3k) ≥ 1-1/k²
P(|X-68.5| < 5.3k) ≥ 0.89
[tex]1 - \frac{1}{k^{2} } = 0.89[/tex]
[tex]\frac{1}{k^{2} } = 0.11\\ \\\frac{1}{k^{2} } = \frac{1}{9} \\\\k = 3[/tex]
Between the following two life expectancies does Chebyshev's Theorem guarantee that we will find at least 89% of smoking males:
= [μ - 3σ, μ + 3σ]
= [68.5 - 3*5.3, 68.5 + 3*5.3]
= [52.6, 84.4]
Learn more about Chebyshev's inequality here
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