Question

the mean value of land and buildings per acre from a sample of farms is $1700 with a standard deviation of $300. the data - set has a bell - shaped distribution. assume the number of farms in the sample is 74.
(a) use the empirical rule to estimate the number of farms whose land and building values per acre are between $1400 and $2000.
(b) if 30 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 $2000 per acre?

the mean value of land and buildings per acre from a sample of farms is $1200 with a standard deviation of $200. the data - set has a bell - shaped distribution. using the empirical rule, determine which of the following farms, whose land and building values per acre are given, are unusual (more than two standard deviations from the mean). are any of the data values very unusual (more than three standard deviations from the mean)?

Explanation:

Step1: Recall empirical rule

For a bell - shaped distribution, about 68% of the data lies within 1 standard deviation of the mean, about 95% lies within 2 standard deviations of the mean, and about 99.7% lies within 3 standard deviations of the mean. The mean $\mu = 1700$ and the standard deviation $\sigma=300$.

Step2: Calculate the number of standard deviations

For the lower bound $x_1 = 1400$, $z_1=\frac{1400 - 1700}{300}=\frac{- 300}{300}=-1$. For the upper bound $x_2 = 2000$, $z_2=\frac{2000 - 1700}{300}=\frac{300}{300}=1$. So the values 1400 and 2000 are 1 standard - deviation below and above the mean respectively.

Step3: Use the empirical rule for part (a)

Since about 68% of the data lies within 1 standard deviation of the mean, and the sample size $n = 74$. The number of farms is $0.68\times74 = 50.32\approx50$.

Step4: Use the empirical rule for part (b)

If 30 additional farms are sampled, and the proportion of data within 1 standard deviation of the mean is 0.68, the expected number of farms is $0.68\times30=20.4\approx20$.

Answer:

a. 50
b. 20