The heights of the adults in one town have a mean of 67.1 inches and a standard deviation of 3.5 inches. What can you conclude from Chebyshev's theorem about the percentage of adults in the town whose heights are between 60.1 and 74.1 inches?
A) The percentage is at most 75% .
B) The percentage is at most 95% .
C) The percentage is at least 95% .
D) The percentage is at least 75% .

Question

The heights of the adults in one town have a mean of  67.1  inches and a standard deviation of  3.5  inches. What can you conclude from Chebyshev's theorem about the percentage of adults in the town whose heights are between  60.1  and  74.1  inches?
A) The percentage is at most  75% . 
B) The percentage is at most  95% .
C) The percentage is at least  95% .  
D) The percentage is at least  75% .

Quizplus · Accepted Answer

Chebyshev's theorem states that for any dataset (regardless of distribution shape), at least $1 - \frac{1}{k^2}$ of the data values will fall within $k$ standard deviations from the mean. Here, $60.1$ and $74.1$ inches are $2$ standard deviations away from the mean ($67.1 \pm 2 \times 3.5$). Thus, $k = 2$, and at least $1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4} = 75\%$ of the data falls within this range.

The Heights of the Adults in One Town Have a Mean