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Chapter 3: Problem 72

Briefly explain Chebyshev's theorem and its applications.

Short Answer

Expert verified
Chebyshev's theorem, or inequality, provides a minimum proportion of data that lies within k standard deviations from the mean of the distribution for any \( k > 1 \). This theorem applies to any distribution, providing a conservative estimate, often used in descriptive statistics, finance and quality control.

Step by step solution

01

Understand Chebyshev's Theorem

Chebyshev's theorem, also known as Chebyshev's inequality, is a concept in probability theory and statistics which states that for a wide variety of data distributions, no more than 1/k^2 of the distribution's values can be more than k standard deviations away from the mean. The theorem is applicable for k>1.
02

Statement of Chebyshev's Theorem

Chebyshev's theorem formally states that: For any \( k > 1 \), at least \(1-1/k^2 \) of data from a dataset will fall within \( k \) standard deviations of the mean. This theorem applies to any distribution regardless of its shape, providing a 'worst-case' analysis of variance.
03

Chebyshev's Theorem Application

Chebyshev's theorem is widely used in descriptive statistics. It acts as a guiding rule in understanding the dispersion of data in a dataset. For instance, for any given dataset, according to this theorem, at least three-fourths \( (1- 1/2^2) \) of data will lie within two (2) standard deviations from the mean. In finance, it can give an estimate of the time a portfolio's returns will deviate from its mean return. In quality control, it gives an estimate of the proportion of products that will fall within certain specifications.

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Most popular questions from this chapter

The following data give the numbers of new cars sold at a dealership during a 20 -day period. \(\begin{array}{lrlrlllll}8 & 5 & 12 & 3 & 9 & 10 & 6 & 12 & 8 \\\ 4 & 16 & 10 & 11 & 7 & 7 & 3 & 5 & 9\end{array}\) a. Calculate the values of the three quartiles and the interquartile range. Where does the value of 4 lie in relation to these quartiles? b. Find the (approximate) value of the 25 th percentile. Give a brief interpretation of this percentile. c. Find the percentile rank of 10 . Give a brief interpretation of this percentile rank.

The following data give the lengths of time (in weeks) taken to find a full- time job by 18 computer science majors who graduated in 2008 from a small college. \(\begin{array}{rrrrrrrrr}30 & 43 & 32 & 21 & 65 & 8 & 4 & 18 & 16 \\ 38 & 9 & 44 & 33 & 23 & 24 & 81 & 42 & 55\end{array}\) Make a box-and-whisker plot. Comment on the skewness of this data set. Does this data set contain any outliers?

The 2009 gross sales of all companies in a large city have a mean of \(\$ 2.3\) million and a standard deviation of \(\$ .6\) million. Using Chebyshev's theorem, find at least what percentage of companies in this city had 2009 gross sales of a. \(\$ 1.1\) to \(\$ 3.5\) million b. \(\$ .8\) to \(\$ 3.8\) million c. \(\$ .5\) to \(\$ 4,1\) million

The following data give the prices (in thousands of dollars) of 20 houses sold recently in a city. \(\begin{array}{llllllllll}184 & 297 & 365 & 309 & 245 & 387 & 369 & 438 & 195 & 390 \\ 323 & 578 & 410 & 679 & 307 & 271 & 457 & 795 & 259 & 590\end{array}\) Find the \(20 \%\) trimmed mean for this data set.

Briefly describe how the three quartiles are calculated for a data set. Illustrate by calculating the three quartiles for two examples, the first with an odd number of observations and the second with an even number of observations.
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