Question

using the empirical rule: the mean value of land and buildings per acre from a sample of farms is $300, with a standard deviation of $50. estimate the percent of farms whose land and building values per acre are between $200 and $400. choose chebychevs theorem: if k = 4, find the percent of data values within 4 standard deviations from the mean. choose true or false: the empirical rule applies to data that is skewed. true false back submit page 3 of 3 clear form never submit passwords through google forms this form was created inside of sealy independent school district does this form look suspicious? report

Explanation:

Step1: Apply the Empirical Rule

The Empirical Rule states that for a normal distribution:

• Approximately \(68\%\) of the data lies within \(1\) standard deviation (\(\mu\pm\sigma\)) of the mean.
• Approximately \(95\%\) of the data lies within \(2\) standard deviations (\(\mu\pm2\sigma\)) of the mean.
• Approximately \(99.7\%\) of the data lies within \(3\) standard deviations (\(\mu\pm3\sigma\)) of the mean.

Given \(\mu = 300\) and \(\sigma=50\).

For \(x_1 = 200\), calculate \(z_1=\frac{200 - 300}{50}=\frac{- 100}{50}=-2\)

For \(x_2 = 400\), calculate \(z_2=\frac{400 - 300}{50}=\frac{100}{50}=2\)

Since \(200 = 300-2\times50\) and \(400 = 300 + 2\times50\), by the Empirical Rule, the percentage of data within \(z=-2\) and \(z = 2\) (i.e., within \(2\) standard deviations of the mean) is \(95\%\)

Step2: Apply Chebyshev's Theorem

Chebyshev's Theorem formula is \(P(|X-\mu|\leq k\sigma)\geq1-\frac{1}{k^{2}}\)

Given \(k = 4\), substitute \(k\) into the formula:

\(P(|X-\mu|\leq4\sigma)\geq1-\frac{1}{4^{2}}=1-\frac{1}{16}=\frac{16 - 1}{16}=\frac{15}{16}=0.9375 = 93.75\%\)

Step3: Determine True - False for Empirical Rule

The Empirical Rule applies to data that is approximately normally distributed (symmetric). Skewed data is not symmetric. So, the statement "The Empirical Rule applies to data that is skewed" is False

Answer:

1. \(95\%\)
2. \(93.75\%\)
3. False