Questions: 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 72. (a) Use the empirical rule to estimate the number of farms whose land and building values per acre are between 1400 and 1600. farms (Round to the nearest whole number as needed.) (b) If 23 additional farms were sampled, about how many of these additional farms would you expect to have land and building values between 1400 per acre and 1600 per acre? farms out of 23 (Round to the nearest whole number as needed.)

Solution Steps

To find the probability of farms whose land and building values per acre are between \$1400 and \$1600, we first calculate the Z-scores for the given values:

\[ Z_{start} = \frac{1400 - 1500}{100} = -1.0 \] \[ Z_{end} = \frac{1600 - 1500}{100} = 1.0 \]

Using the standard normal distribution, we find:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.0) - \Phi(-1.0) = 0.6827 \]

Given that the total number of farms in the sample is 72, we can estimate the number of farms within the specified range:

\[ \text{Estimated Farms} = P \times \text{Total Farms} = 0.6827 \times 72 \approx 49 \]

If 23 additional farms are sampled, we can estimate how many of these would also fall within the same range:

\[ \text{Expected Additional Farms} = P \times \text{Additional Farms} = 0.6827 \times 23 \approx 16 \]

Final Answer