Question

The mean value of land and buildings per acre from a sample of farms is ​$1200​, with a standard deviation of ​$100. The data set has a​ bell-shaped distribution. Using the empirical​ rule, determine which of the following​ farms, whose land and building values per acre are​ given, are unusual​ (more than two standard deviations from the​ mean). Are any of the data values very unusual​ (more than three standard deviations from the​ mean)? ​$1064 ​$1417 ​$1373 ​$856 ​$1250 ​$1102

Answer 1

The empirical rule states that for a bell-shaped distribution, approximately 68% of the data values will fall within one standard deviation of the mean, 95% of the data values will fall within two standard deviations of the mean, and 99.7% of the data values will fall within three standard deviations of the mean.

Using this information, we can calculate the range of values that fall within two standard deviations of the mean for the given data set:

Mean: $1200

Standard deviation: $100

Two standard deviations: $200

Therefore, the range of values that fall within two standard deviations of the mean is $1200 - $200 = $1000 to $1200 + $200 = $1400.

With this information, we can determine which of the given data values are unusual (more than two standard deviations from the mean):

$1064 is within two standard deviations of the mean.

$1417 is outside of two standard deviations of the mean.

$1373 is within two standard deviations of the mean.

$856 is outside of two standard deviations of the mean.

$1250 is within two standard deviations of the mean.

$1102 is within two standard deviations of the mean.

Therefore, the data values $1417 and $856 are unusual (more than two standard deviations from the mean).

We can also determine which of the given data values are very unusual (more than three standard deviations from the mean) using the same process. The range of values that fall within three standard deviations of the mean is $1000 to $1600. With this information, we can see that none of the given data values are very unusual (more than three standard deviations from the mean).