Question

discuss the similarities and the differences between the empirical rule and chebychevs theorem. what is a similarity between the empirical rule and chebychevs theorem? a. both calculate the variance and standard deviation of a sample. b. both do not require the data to have a sample standard deviation. c. both estimate proportions of the data contained within k standard deviations of the mean. d. both apply only to symmetric and bell - shaped distributions.

Explanation:

• Empirical Rule: Applies to symmetric, bell - shaped (normal) distributions. It gives specific proportions of data within \(k = 1,2,3\) standard deviations of the mean (\(68\%\) for \(k = 1\), \(95\%\) for \(k=2\), \(99.7\%\) for \(k = 3\)).
• Chebychev's Theorem: Applies to any distribution (regardless of shape). The formula is \(P(|X-\mu|\leq k\sigma)\geq1-\frac{1}{k^{2}}\) (\(k> 1\)), which estimates the proportion of data within \(k\) standard deviations of the mean.
• Option A: Incorrect. Neither the Empirical Rule nor Chebychev's Theorem calculates variance and standard deviation. These are measures that might be used as inputs (if known) for the rules/theorems.
• Option B: Incorrect. Both rules/theorems rely on the concept of standard deviation (either population \(\sigma\) or sample \(s\)).
• Option C: Correct. The Empirical Rule gives exact proportions (\(0.68,0.95,0.997\)) for \(k = 1,2,3\) in a normal distribution, and Chebychev's Theorem gives a lower - bound proportion (\(1-\frac{1}{k^{2}}\)) for any \(k>1\) in any distribution.
• Option D: Incorrect. The Empirical Rule applies to symmetric and bell - shaped distributions, while Chebychev's Theorem applies to any distribution.

Answer:

C. Both estimate proportions of the data contained within \(k\) standard deviations of the mean.