Q:

A statistician uses Chebyshev's Theorem to estimate that at least 15 % of a population lies between the values 9 and 20. Use this information to find the values of the population mean, μ , and the population standard deviation σ.

Accepted Solution

A:
Answer:\mu = 14.5\\ \sigma = 5.071\\ k = 1.084Step-by-step explanation:given that a  statistician uses Chebyshev's Theorem to estimate that at least 15 % of a population lies between the values 9 and 20. i.e. his findings with respect to probability are[tex]P(9<x<20) \geq 0.15\\P(|x-14.5|<5.5) \geq 0.15[/tex]Recall Chebyshev's inequality that[tex]P(|X-\mu |\geq k\sigma )\leq {\frac {1}{k^{2}}}\\P(|X-\mu |\leq k\sigma )\geq 1-{\frac {1}{k^{2}}}\\[/tex]Comparing with the Ii equation which is appropriate here we find that[tex]\mu =14.5[/tex]Next what we find is[tex]k\sigma = 5.5\\1-\frac{1}{k^2} =0.15\\\frac{1}{k^2}=0.85\\k=1.084\\1.084 (\sigma) = 5.5\\\sigma = 5.071[/tex]Thus from the given information we find that[tex]\mu = 14.5\\\sigma = 5.071\\k = 1.084[/tex]