Questions: The mean value of land and buildings per acre from a sample of farms is 1500, with a standard deviation of 200. The data set has a bell-shaped distribution. Assume the number of farms in the sample is 75. (a) Use the empirical rule to estimate the number of farms whose land and building values per acre are between 1100 and 1900. farms (Round to the nearest whole number as needed.)

The mean value of land and buildings per acre from a sample of farms is 1500, with a standard deviation of 200. The data set has a bell-shaped distribution. Assume the number of farms in the sample is 75.

(a) Use the empirical rule to estimate the number of farms whose land and building values per acre are between 1100 and 1900.

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

Transcript text: The mean value of land and buildings per acre from a sample of farms is $\$ 1500$, with a standard deviation of $\$ 200$. The data set has a bell-shaped distribution. Assume the number of farms in the sample is 75. (a) Use the empirical rule to estimate the number of farms whose land and building values per acre are between $\$ 1100$ and $\$ 1900$. $\square$ farms (Round to the nearest whole number as needed.)

Solution

Solution Steps

Step 1: Calculate Z-scores

To find the Z-scores for the values $1100$ and $1900$, we use the formula:

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

For $X = 1100$:

\[ z_{1100} = \frac{1100 - 1500}{200} = -2.0 \]

For $X = 1900$:

\[ z_{1900} = \frac{1900 - 1500}{200} = 2.0 \]

Step 2: Calculate Probability

Using the Z-scores, we can find the probability that the values fall between $1100$ and $1900$:

\[ P = \Phi(z_{1900}) - \Phi(z_{1100}) = \Phi(2.0) - \Phi(-2.0) = 0.9545 \]

Step 3: Estimate Number of Farms

To estimate the number of farms whose land and building values per acre are between $1100$ and $1900$, we multiply the probability by the sample size:

\[ \text{Estimated Farms} = P \times n = 0.9545 \times 75 \approx 72 \]