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The mean value of land and buildings per acre from a sample of farms is $1800, with a standard deviation of $100. The data set has a bell-shaped distribution. Assume the number of farms in the sample is 77. (a) Use the empirical rule to estimate the number of farms whose land and building values per acre are between $1600 and $2000. farms (Round to the nearest whole number as needed.)

The mean value of land and buildings per acre from a sample of farms is $1800, with a standard deviation of $100. The data set has a bell-shaped distribution. Assume the number of farms in the sample is 77. (a) Use the empirical rule to estimate the number of farms whose land and building values per acre are between $1600 and $2000. farms (Round to the nearest whole number as needed.)

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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### Estimating Farm Values Using the Empirical Rule **Problem Statement:** The mean value of land and buildings per acre from a sample of farms is $1800, with a standard deviation of $100. The data set has a bell-shaped distribution. Assume the number of farms in the sample is 77. --- (a) **Use the empirical rule to estimate the number of farms whose land and building values per acre are between \$1600 and \$2000.** #### Steps to Solve: 1. **Determine the Z-scores:** First, calculate how many standard deviations away the values \$1600 and \$2000 are from the mean value of \$1800. - \$1600 is (1800 - 1600) / 100 = 2 standard deviations below the mean. - \$2000 is (2000 - 1800) / 100 = 2 standard deviations above the mean. 2. **Apply the Empirical Rule:** - According to the empirical rule for a normal distribution: - Approximately 68% of data falls within ±1 standard deviation of the mean. - Approximately 95% of data falls within ±2 standard deviations of the mean. - Approximately 99.7% of data falls within ±3 standard deviations of the mean. Since \$1600 to \$2000 is within ±2 standard deviations from the mean, approximately 95% of the data lies within this range. 3. **Calculate the Number of Farms:** - 95% of 77 farms = 0.95 * 77 - Number of farms ≈ 73.15 Round to the nearest whole number. **Answer:** There are approximately **73 farms** whose land and building values per acre are between \$1600 and \$2000. __*(Round to the nearest whole number as needed.)*__
Transcribed Image Text:### Estimating Farm Values Using the Empirical Rule **Problem Statement:** The mean value of land and buildings per acre from a sample of farms is $1800, with a standard deviation of $100. The data set has a bell-shaped distribution. Assume the number of farms in the sample is 77. --- (a) **Use the empirical rule to estimate the number of farms whose land and building values per acre are between \$1600 and \$2000.** #### Steps to Solve: 1. **Determine the Z-scores:** First, calculate how many standard deviations away the values \$1600 and \$2000 are from the mean value of \$1800. - \$1600 is (1800 - 1600) / 100 = 2 standard deviations below the mean. - \$2000 is (2000 - 1800) / 100 = 2 standard deviations above the mean. 2. **Apply the Empirical Rule:** - According to the empirical rule for a normal distribution: - Approximately 68% of data falls within ±1 standard deviation of the mean. - Approximately 95% of data falls within ±2 standard deviations of the mean. - Approximately 99.7% of data falls within ±3 standard deviations of the mean. Since \$1600 to \$2000 is within ±2 standard deviations from the mean, approximately 95% of the data lies within this range. 3. **Calculate the Number of Farms:** - 95% of 77 farms = 0.95 * 77 - Number of farms ≈ 73.15 Round to the nearest whole number. **Answer:** There are approximately **73 farms** whose land and building values per acre are between \$1600 and \$2000. __*(Round to the nearest whole number as needed.)*__
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