Question

the mean value of land and buildings per acre from a sample of farms is $1700, with a standard deviation of $200. the data set has a bell - shaped distribution. assume the number of farms in the sample is 85.

1. about 68% of the data lie within one standard deviation of the mean.
2. about 95% of the data lie within two standard deviations of the mean.
3. about 99.7% of the data lie within three standard deviations of the mean.

sometimes, the more precise percentages 68.26%, 95.44%, and 99.72% are used. in this exercise, however, use 68%, 95%, and 99.7% as appropriate.
(a) to find the number of farms whose land and building values per acre are between $1500 and $1900, find how many standard deviations lie between the mean value $1700 and the border values $1500 or $1900. there is one standard deviation that lies between the mean value and a border value.
according to empirical rule, there are about 68% of the data that lie within one standard deviation from the mean.
the number of farms in the sample is equal 85. hence, 68% of the number of farms in the sample is approximately 58
so, there are approximately 58 farms whose land and building values per acre are between $1500 and $1900.

Explanation:

Step1: Identify the percentage of data within one standard deviation

According to the empirical rule, about 68% of the data lies within one standard deviation of the mean.

Step2: Calculate the number of farms

The total number of farms in the sample is \(n = 85\). To find the number of farms within one - standard deviation, we use the formula \(N=n\times p\), where \(p = 0.68\) (the proportion of data within one standard deviation). So \(N=85\times0.68\).

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Answer:

There are approximately 58 farms whose land and building values per acre are between the mean value minus one standard deviation and the mean value plus one standard deviation.