Question

The mean value of land and buildings per acre from a sample of farms is $1500 , with a standard deviation of $100. The data set has a bell-shaped distribution. Assume the number of farms in the sample is 71. (a) Use the empirical rule to estimate the number of farms whose land and building values per acre are between $1300 and $1700

Answer #1

Emperical Rule ( 68 - 95 - 99.7) states that,

Approximately 68% data falls in 1 standard deviation of the mean.

Approximately 95% data falls in 2 standard deviation of the mean.

Approximately 99.7% data falls in 3 standard deviation of the mean.

Given = 1500, = 100

We have to find P( 1300 < X < 1700) = ?

1300 = 1500 - 2 * 100

That is

1300 = - 2 *

That is 1300 is 2 standard deviation below the mean.

Similarly,

1700 = + 2 *

That is 1700 is 2 standard deviation above the mean.

So, 1300 and 1700 are 2 standard deviation below and above the mean, that is 2 standard deviation of the mean.

By emperical rule, approximately 95% of the data falls in the 2 standard deviation of the mean.

P( 1300 < X < 1700) = 0.95

So in 71 samples, number of farms falls between $1300 and $1700 is

0.95 * 71 = 0.6745

= **68** (Rounded to nearest integer)

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