Question

the mean value of land and buildings per acre from a sample of farms is $1800, with a standard deviation of $200. the data set has a bell - shaped distribution. assume the number of farms in the sample is 76.
(a) use the empirical rule to estimate the number of farms whose land and building values per acre are between $1400 and $2200.
72 farms (round to the nearest whole number as needed.)
(b) if 24 additional farms were sampled, about how many of these additional farms would you expect to have land and building values between $1400 per acre and $2200 per acre?
22 farms out of 24 (round to the nearest whole number as needed.)

Explanation:

Step1: Calculate the number of standard deviations from the mean

For \( \$1400\): \(z_1=\frac{1400 - 1800}{200}=\frac{- 400}{200}=-2\)
For \( \$2200\): \(z_2=\frac{2200 - 1800}{200}=\frac{400}{200}=2\)

Step2: Apply the empirical rule

The empirical rule states that for a bell - shaped (normal) distribution, approximately \(95\%\) of the data lies within \(z=-2\) and \(z = 2\)

Step3: Calculate the number of farms in part (a)

If the sample size \(n = 76\), the number of farms is \(0.95\times76=72.2\approx72\)

Step4: Calculate the number of farms in part (b)

If \(n = 24\), the number of farms is \(0.95\times24 = 22.8\approx23\)

Answer:

(a) \(72\) farms
(b) \(23\) farms