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Chapter 3: Problem 72

Briefly explain Chebyshev's theorem and its applications.

Short Answer

Expert verified
Chebyshev's theorem, or Chebyshev's inequality, gives an approximation of the minimum amount of data within k standard deviations from the mean for any dataset. The theorem is applicable to any dataset and is used widely in processes like quality control and finance to predict variability or defects.

Step by step solution

01

Explain Chebyshev's Theorem

Chebyshev鈥檚 theorem, also known as Chebyshev鈥檚 inequality, is a probabilistic theorem that provides an approximation to the bare minimum proportion of data values that fall within k standard deviations from the mean for any given dataset, regardless of the shape of the data distribution. The formula for Chebyshev's theorem is \(1 - 1/k^2\) where k is the number of standard deviations from the mean.
02

Example of Chebyshev's Theorem

For instance, considering k is 2 (number of standard deviations from the mean), substituting the value of k in Chebyshev's theorem formula, we get \((1 - 1/k^2) = 1 - 1/2^2 = 0.75\). This implies at least 75% of data falls within 2 standard deviations from the mean in any data distribution.
03

Application of Chebyshev's Theorem

Chebyshev's theorem is broad-spectrum, meaning it鈥檚 applicable to any dataset, whether it鈥檚 normal distribution or not. In statistics, it is useful for making estimations about the proportion of data, which falls within a certain number of standard deviations from the mean. For instance, quality control processes often utilize Chebyshev鈥檚 theorem to predict the number of defective items that will be included in a shipment or delivery. It is also used in finance to understand the variability of investment returns.

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Most popular questions from this chapter

According to www.money-zine.com, the average FICO score in the United States was around 692 in December \(2011 .\) Suppose the following data represent the credit scores of 22 randomly selected loan applicants. \(\begin{array}{lllllllllll}494 & 728 & 468 & 533 & 747 & 639 & 430 & 690 & 604 & 422 & 356 \\ 805 & 749 & 600 & 797 & 702 & 628 & 625 & 617 & 647 & 772 & 572\end{array}\) a. Calculate the values of the three quartiles and the interquartile range. Where does the value 617 fall in relation to these quartiles? b. Find the approximate value of the 30 th percentile. Give a brief interpretation of this percentile. c. Calculate the percentile rank of 533 . Give a brief interpretation of this percentile rank.

Consider the following two data sets. \(\begin{array}{llllrl}\text { Data Set 1: } & 12 & 25 & 37 & 8 & 41 \\ \text { Data Set II: } & 19 & 32 & 44 & 15 & 48\end{array}\) Notice that each value of the second data set is obtained by adding 7 to the corresponding value of the first data set. Calculate the mean for each of these two data sets. Comment on the relationship between the two means.

Refer to the data in Exercise 3.23, which contained the numbers of tornadoes that touched down in 12 states that had the most tornadoes during the period 1950 to 1994 . The data are reproduced here. \(\begin{array}{lllllllll}1113 & 2009 & 1374 & 1137 & 2110 & 1086 & 1166 & 1039 & 1673 & 2300\end{array}\) \(\begin{array}{ll}1139 & 5490\end{array}\) Find the range, variance, and standard deviation for these data.

Refer to the data of Exercise \(3.109\) on the current annual incomes (in thousands of dollars) of the 10 members of the class of 2004 of the Metro Business College who were voted most likely to succeed. \(\begin{array}{lllllllll}59 & 68 & 84 & 78 & 107 & 382 & 56 & 74 & 97 & 60\end{array}\) a. Determine the values of the three quartiles and the interquartile range. Where does the value of 74 fall in relation to these quartiles? b. Calculate the (approximate) value of the 70 th percentile. Give a brief interpretation of this percentile. c. Find the percentile rank of 97 . Give a brief interpretation of this percentile rank.

The following table gives the frequency distribution of the times (in minutes) that 50 commuter students at a large university spent looking for parking spaces on the first day of classes in the Fall semester of 2012 . \begin{tabular}{lc} \hline \multicolumn{1}{c} { Time } & Number of Students \\ \hline 0 to less than 4 & 1 \\ 4 to less than 8 & 7 \\ 8 to less than 12 & 15 \\ 12 to less than 16 & 18 \\ 16 to less than 20 & 6 \\ 20 to less than 24 & 3 \\ \hline \end{tabular} Find the mean, variance, and standard deviation. Are the values of these summary measures population parameters or sample statistics?
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