Answer: 84%
Explanation:
Let
x = any income in the sample observation
[tex] \mu [/tex] = mean = $500
[tex] \sigma [/tex] = standard deviation = $40
k = any positive numbers
Chebyshev's theorem states that at least (1 - 1/k²) of the incomes is within k standard deviations from the mean.
In terms of mathematical equation:
[tex]P(|x - \mu| \leq k\sigma) \geq 1 - \frac{1}{k^2} [/tex]
To use Chebyshev's theorem, we get the expressions for
[tex]|x - \mu| = |x - 500|[/tex]
Since we are concerned with the incomes between $400 and $600,
[tex]400 \leq x \leq 600
\newline \Leftrightarrow -100 \leq x - 500 \leq 100
\newline \Leftrightarrow |x - 500| \leq 100
\newline \Leftrightarrow |x - 500| \leq 2.5(40)[/tex]
Thus, we take k = 2.5. By Chebyshev's theorem,
[tex]P(|x - \mu| \leq k\sigma) \geq 1 - \frac{1}{k^2}
\newline P(|x - 500| \leq 2.5(40)) \geq 1 - \frac{1}{2.5^2}
\newline P(|x - 500| \leq 100) \geq 1 - 0.16
\newline P(-100 \leq x - 500 \leq 100) \geq 0.84
\newline \boxed{P(400 \leq x \leq 600) \geq 0.84} [/tex]
Therefore, at least 84% of the incomes will lie between $400 and $600.
