Using Chebyshev’s Theorem

Chebyshev’s theorem states that for any distribution (not necessarily normal), the proportion of values that lie within $k$ standard deviations of the mean is at least:

$1 - \frac{1}{k^2}$

where $k$ is the number of standard deviations from the mean.

Given Data

• Mean $\mu = 20$
• Standard Deviation $\sigma = 2$

Part (a): Percentage of values between 10 and 30

1. Find $k$:

   • The lower bound is 10, and the upper bound is 30.
   • Distance from the mean:
     $k = \frac{|10 - 20|}{2} = \frac{10}{2} = 5$ (or $k = \frac{|30 - 20|}{2} = 5$, since the range is symmetric).

2. Apply Chebyshev’s theorem:
   $1 - \frac{1}{5^2} = 1 - \frac{1}{25} = 1 - 0.04 = 0.96$

   At least 96% of the values fall between 10 and 30.

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Part (b): Percentage of values between 12 and 28

1. Find $k$:

   • The lower bound is 12, and the upper bound is 28.
   • Distance from the mean:
     $k = \frac{|12 - 20|}{2} = \frac{8}{2} = 4$ (or $k = \frac{|28 - 20|}{2} = 4$, since the range is symmetric).

2. Apply Chebyshev’s theorem:
   $1 - \frac{1}{4^2} = 1 - \frac{1}{16} = 1 - 0.0625 = 0.9375$

   At least 93.75% of the values fall between 12 and 28.

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Final Answers:

(a) At least 96% of the values fall between 10 and 30.
(b) At least 93.75% of the values fall between 12 and 28.

Would you like a deeper explanation or another example?

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Related Questions:

1. What is Chebyshev’s theorem, and why is it useful for non-normal distributions?
2. How does Chebyshev’s theorem compare to the empirical rule for normal distributions?
3. If the standard deviation were 3 instead of 2, how would that change the results?
4. Can Chebyshev’s theorem be applied to skewed distributions?
5. What happens if $k = 1$ in Chebyshev’s theorem?

Tip: Chebyshev’s theorem is most useful when dealing with unknown or non-normal distributions, ensuring a minimum proportion of data falls within a given range.