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
□ farms (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 \(P(-1<Z<1)=0.68\)

Step3: Calculate the number of farms

The number of farms in the sample is \(n = 74\).
The number of farms with values between \(1200\) and \(1600\) is \(n\times P(-1 < Z<1)=74\times0.68 = 50.32\approx50\)

Answer:

\(50\)