The mean value of land and buildings per acre from a sample of farms is $1400, with a standard deviation of $200. The data set has a bell-shaped distribution. Assume the number of farms in the sample is 74. (a) Use the empirical rule to estimate the number of farms whose land and building values per acre are between $1200 and $1600.
(b) If 29 additional farms were sampled, about how many of these additional farms would you expect to have land and building values between $1200 per acre and $1600 per acre?

The answer is below

Explanation:

The Empirical Rule (or 3 sigma rule) states that for a normal distribution (bell shaped distribution) 68% of the data falls within one standard deviation (μ ± σ), 95% percent within two standard deviations (μ ± 2σ), and 99.7% within three standard deviations from the mean (μ ± 3σ).

Given that the mean (μ) = $1400, standard deviation (σ) = $200

a) The percentage of data within one standard deviation = μ ± σ = (1400 ± 200) = (1200, 1600)

Hence 68% of the land are between $1200 and $1600.

Number of farms = 68% × number of sample = 0.68 × 74 = 50.23 ≈ 51 farms

b) For an additional 29 farms, the number of additional farms between $1200 per acre and $1600 per acre = 29 × 0.68 ≈ 20 farms