Question

The average farm in the united states in 2014 contained 504 acres. The standard deviation is 55.7 acres. Use Chebyshev's theorem to find the minimum percentage of data values that will fall in the range of 392.5 and 896.57 acres.

Answer 1

According to Chebyshev's inequality theorem, at least 1/k² of the total values lies within k standard deviations of the mean.

392.5 = 504 - 111.5 which is 2 standard deviations less than the mean.
896.57 = 504 + 7*55.7 which is 7 standard deviations above mean.

According to Chebyshev's inequality, at least 1/2² = 25% of the observations lies outside 2 standard deviations of the mean.
Also, at least 1/7² = 2.04% of the observations lies outside 7 standard deviations from the mean.

This means that maximum (2.04)/2 + 25/2 = 13.52% of the observations lies outside the given interval. Therefore 1 - 13.52% = 86.48% of the observations lies between 392.5 and 896.57