Questions: 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 71. (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.)

Solution Steps

To determine the number of farms whose land and building values per acre are between \$900 and \$1500, we first calculate the Z-scores for these values using the formula:

\[ Z = \frac{X - \mu}{\sigma} \]

where:

• \(X\) is the value,
• \(\mu = 1200\) (mean),
• \(\sigma = 300\) (standard deviation).

Calculating for \$900:

\[ Z_{900} = \frac{900 - 1200}{300} = -1.0 \]

Calculating for \$1500:

\[ Z_{1500} = \frac{1500 - 1200}{300} = 1.0 \]

According to the empirical rule (68-95-99.7 rule), approximately 68% of the data in a normal distribution falls within one standard deviation of the mean. Since the Z-scores for \$900 and \$1500 are \(-1.0\) and \(1.0\) respectively, this range encompasses about 68% of the farms.

Given that the sample size is \(n = 71\), we can estimate the number of farms within the range of \$900 to \$1500:

\[ \text{Number of farms} = 0.68 \times n = 0.68 \times 71 \approx 48.28 \]

Rounding to the nearest whole number gives us:

\[ \text{Estimated number of farms} = 48 \]

Final Answer