The mean value of land and buildings per acre from a sample

The mean value of land and buildings per acre from a sample of farms is $1600,with a standard deviation of $300. The data set has a bell-shaped distribution. Assume the number of farms in the sample is 77.
a) Use the empirical rule to estimate the number of farms whose land and building values per acre are between $1300 and $1900.

? farms(Round to the nearest whole number as needed.)

Solution

Note that the distribution need to be both bell-shaped and symmetrical before you can use the empirical rule.

(a)

mean = 1600
Std Dev = 300

Let X = value of lands and buildings of farm per acre
P(1300 < X < 1900)
= P(((1300 - 1600)/300) < Z < ((1900 - 1600)/300))
= P(-1 < Z < 1)

From the empirical rule, 68% of the farms have lands and buildings which are between $1400 and $2000 per acre.