Question

the mean value of land and buildings per acre from a sample of farms is $1500, 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 $1100 and $1900. farms (round to the nearest whole number as needed)

Explanation:

Step1: Calculate number of standard - deviations from the mean

First, find how many standard - deviations $1100$ and $1900$ are from the mean $\mu = 1500$ with standard deviation $\sigma=200$.
For $x_1 = 1100$: $z_1=\frac{1100 - 1500}{200}=\frac{- 400}{200}=-2$.
For $x_2 = 1900$: $z_2=\frac{1900 - 1500}{200}=\frac{400}{200}=2$.

Step2: Apply the empirical rule

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

Step3: Calculate the number of farms

We have a sample of $n = 74$ farms. The number of farms within the range is $0.95\times n$.
$0.95\times74 = 70.3\approx70$.

Answer:

$70$