Question

farm sizes the average farm in the united states in a certain year contained 438 acres. the standard deviation is 24 acres. use chebyshevs theorem to find the minimum percentage of data values that will fall in the range of 354 - 522 acres. round your answer to the nearest percentage. minimum percentage of data values that will fall in the range of 354 - 522 acres is \square %.

Explanation:

Step1: Calculate k value

First, find how many standard deviations the range is from the mean.
Mean $\mu = 438$, standard deviation $\sigma = 24$.
Lower bound: $438 - 354 = 84$; Upper bound: $522 - 438 = 84$
$k = \frac{84}{24} = 3.5$

Step2: Apply Chebyshev's Theorem

Chebyshev's formula: $1 - \frac{1}{k^2}$
Substitute $k=3.5$:
$1 - \frac{1}{(3.5)^2} = 1 - \frac{1}{12.25}$

Step3: Compute the percentage

Calculate the value:
$1 - \frac{1}{12.25} = 1 - 0.0816 \approx 0.9184$
Convert to percentage: $0.9184 \times 100 = 91.84\%$

Step4: Round to nearest percentage

Round $91.84\%$ to the nearest whole number.

Answer:

92%