Question

farm sizes the average farm in the united states in a certain year contained 436 acres. the standard deviation is 28 acres. use chebyshevs theorem to find the minimum percentage of data values that will fall in the range of 338 - 534 acres. round your answer to the nearest percentage.
minimum percentage of data values that will fall in the range of 338 - 534 acres is
%

Explanation:

Step1: Find the number of standard deviations (k)

The mean ($\mu$) is 436 acres, and the standard deviation ($\sigma$) is 28 acres. For the lower bound 338: $436 - 338 = 98$. For the upper bound 534: $534 - 436 = 98$. So, $k\sigma = 98$. Since $\sigma = 28$, we solve for $k$: $k=\frac{98}{28}=3$.

Step2: Apply Chebyshev's theorem

Chebyshev's theorem states that the minimum percentage of data within $k$ standard deviations of the mean is $1 - \frac{1}{k^2}$. Substituting $k = 3$: $1-\frac{1}{3^2}=1 - \frac{1}{9}=\frac{8}{9}\approx0.8889$. Convert to percentage: $0.8889\times100\approx89\%$.

Answer:

89