Question

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. 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)?

\\$1698 \\$1772 \\$1494 \\$1197 \\$1618 \\$1351

which of the farms are unusual (more than two standard deviations from the mean)? select all that apply.

a. \\$1197
b. \\$1494
c. \\$1698
d. \\$1772
e. \\$1618
f. \\$1351

which of the farms are very unusual (more than three standard deviations from the mean)? select all that apply.

a. \\$1351
b. \\$1772
c. \\$1494
d. \\$1698
e. \\$1197
f. \\$1618
g. none of the data values are very unusual.

Explanation:

Identify parameters and thresholds

Using the Empirical Rule Percentiles and Outlier Detection knowledge points
Given:
Mean \(\mu = 1500\)
Standard deviation \(\sigma = 100\)

Data values:
\(x_1 = 1698\), \(x_2 = 1772\), \(x_3 = 1494\), \(x_4 = 1197\), \(x_5 = 1618\), \(x_6 = 1351\)

Thresholds for "unusual" (more than \(2\sigma\) from the mean):

$$ \mu \pm 2\sigma = 1500 \pm 2(100) = [1300, 1700] $$

Values are unusual if \(x < 1300\) or \(x > 1700\).

Thresholds for "very unusual" (more than \(3\sigma\) from the mean):

$$ \mu \pm 3\sigma = 1500 \pm 3(100) = [1200, 1800] $$

Values are very unusual if \(x < 1200\) or \(x > 1800\).

Determine unusual values

Using the Outlier Detection knowledge point
We compare each data value to the interval \([1300, 1700]\):

• \(1698\): Within interval (not unusual)
• \(1772\): Outside interval (\(1772 > 1700\), unusual)
• \(1494\): Within interval (not unusual)
• \(1197\): Outside interval (\(1197 < 1300\), unusual)
• \(1618\): Within interval (not unusual)
• \(1351\): Within interval (not unusual)

Thus, the unusual values are \(\$1197\) and \(\$1772\).

Determine very unusual values

Using the Outlier Detection knowledge point
We compare each data value to the interval \([1200, 1800]\):

• \(1698\): Within interval
• \(1772\): Within interval
• \(1494\): Within interval
• \(1197\): Outside interval (\(1197 < 1200\), very unusual)
• \(1618\): Within interval
• \(1351\): Within interval

Thus, the only very unusual value is \(\$1197\).

Answer:

Question 1

• (A) $1197 (Correct answer)
• (B) $1494
• (C) $1698
• (D) $1772 (Correct answer)
• (E) $1618
• (F) $1351

Question 2

• (A) $1351
• (B) $1772
• (C) $1494
• (D) $1698
• (E) $1197 (Correct answer)
• (F) $1618
• (G) None of the data values are very unusual.