Q. 3.107 In this exercise, you will compa... [FREE SOLUTION]

Chapter 3: Q. 3.107 (page 122)

In this exercise, you will compare Chebyshev's rule and the empirical rule.
a. Compare the estimates given by the two rules for the percentage of observations that lie within two standard deviations to either side of the mean. Comment on the differences.
b. Compare the estimates given by the two rules for the percentage of observations that lie within three standard deviations to either side of the mean. Comment on the differences.

Short Answer

Expert verified

a. According to Chebyshev's rule $75 %$ of observations lie within two standard deviations to either side of the mean where as according to empirical rule $95 %$ of observations lie within two standard deviations to either side of the mean

b. According to Chebyshev's rule $88.9 %$ of observations lie within three standard deviations to either side of the mean where as according to empirical rule $99.7 %$ of observations lie within three standard deviations to either side of the mean

Step by step solution

Part(a) Step 1: Given information

We need to compare Chebyshev's rule and empirical rule for

$S \tan d a r d d e v i a t i o n = 2$

Part(a) Step 2: Explanation

Chebyshev's Theorem indicates that at least $75$ percent of observations fall within a range of two standard deviations around the mean, and no more than $25$ percent fall beyond that range, but the empirical rule predicts that $95$ percent fall inside the first two standard deviations.

Chebyshev's theorem differs from the other because it makes no assumptions about the distribution. It is applicable to all forms of data distributions, whereas the Empirical theorem presupposes that data is normally distributed.

Part(b) Step 1: Given information

We need to compare Chebyshev's rule and empirical rule for

$S \tan d a r d d e v i a t i o n = 3$

Part(b) Step 2: Explanation

Chebyshev's Theorem asserts that at least $88.9$ percent of observations fall within a range of three standard deviations around the mean, and no more than $11.1$ percent fall beyond that range, whereas the empirical rule predicts that $99.7$ $%$ fall inside the first three standard deviations.