Question

the mean value of land and buildings per acre from a sample of farms is $1200, with a standard deviation of $300. the data set has a bell - shaped distribution. assume the number of farms in the sample is 70. (a) use the empirical rule to estimate the number of farms whose land and building values per acre are between $900 and $1500. farms (round to the nearest whole number as needed.)

Explanation:

Step1: Calculate z - scores

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu = 1200$ (mean), $\sigma=300$ (standard deviation).
For $x = 900$, $z_1=\frac{900 - 1200}{300}=\frac{- 300}{300}=-1$.
For $x = 1500$, $z_2=\frac{1500 - 1200}{300}=\frac{300}{300}=1$.

Step2: Apply empirical rule

The empirical rule for a normal (bell - shaped) distribution states that approximately 68% of the data lies within 1 standard deviation of the mean, i.e., between $z=-1$ and $z = 1$.

Step3: Calculate number of farms

The number of farms in the sample is $n = 70$.
The number of farms within 1 standard deviation of the mean is $0.68\times n$.
So, $0.68\times70 = 47.6$.
Rounding to the nearest whole number, we get 48.

Answer:

48