Question

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.
50 farms (round to the nearest whole number as needed.)
(b) if 27 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?
farms out of 27 (round to the nearest whole number as needed.)

Explanation:

Step1: Calculate the number of standard deviations from the mean

The mean is \(\mu = 1400\), the standard deviation is \(\sigma=200\).
For \(x = 1200\), \(z=\frac{1200 - 1400}{200}=\frac{- 200}{200}=-1\)
For \(x = 1600\), \(z=\frac{1600 - 1400}{200}=\frac{200}{200}=1\)

Step2: Apply the empirical rule

The empirical rule states that for a bell - shaped (normal) distribution, approximately \(68\%\) of the data lies within \(z=-1\) and \(z = 1\)

Step3: Calculate the expected number of farms in the additional sample

If \(68\%\) of the data lies within the range, and the number of additional farms \(n = 27\)
The expected number \(E=0.68\times27\)

Answer:

\(E = 0.68\times27=18.36\approx18\) farms out of \(27\)