/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Consider sample data with \(\bar... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 15

Consider sample data with \(\bar{x}=15\) and \(s=3\). (a) Compute the coefficient of variation. (b) Compute a \(75 \%\) Chebyshev interval around the sample mean.

Short Answer

Expert verified
(a) Coefficient of Variation is 20%. (b) 75% Chebyshev interval: [9, 21].

Step by step solution

01

Understand the Coefficient of Variation

The coefficient of variation (CV) is a measure of relative variability calculated as the ratio of the standard deviation to the mean. It is expressed as a percentage and is given by the formula: \[ CV = \left( \frac{s}{\bar{x}} \right) \times 100\% \], where \( s \) is the standard deviation and \( \bar{x} \) is the mean.
02

Calculate the Coefficient of Variation

Using the given values \( \bar{x}=15 \) and \( s=3 \), substitute these into the formula to get: \[ CV = \left( \frac{3}{15} \right) \times 100\% = 20\% \]. So, the coefficient of variation is 20%.
03

Understand Chebyshev's Interval

Chebyshev's theorem states that for any data set, the proportion of values within \( k \) standard deviations of the mean is at least \( 1 - \frac{1}{k^2} \). For a specified percentage, solve for \( k \) to determine the interval.
04

Determine k for 75% Chebyshev Interval

We want a 75% interval, so set \( 1 - \frac{1}{k^2} = 0.75 \). Solving for \( k \) gives \( \frac{1}{k^2} = 0.25 \), thus \( k^2 = 4 \), and hence \( k = 2 \).
05

Calculate the 75% Chebyshev Interval

Use \( k = 2 \) to calculate the interval around the mean. The bounds are \( \bar{x} \pm ks = 15 \pm 2 \times 3 = 15 \pm 6 \), i.e., the interval is from 9 to 21.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Coefficient of Variation
The coefficient of variation (CV) is a valuable statistical measure that helps us understand the level of variability in a dataset relative to its mean. It's essentially a standardized measure of dispersion and provides insights into how much variation there is in data compared to the average of that data.
One key feature of CV is that, because it's expressed as a percentage, it allows you to compare the level of variation between datasets with different units or scales.
  • The formula for calculating the coefficient of variation is: \( CV = \left( \frac{s}{\bar{x}} \right) \times 100\% \) where \( s \) is the standard deviation, and \( \bar{x} \) is the mean.
  • For example, given a mean \( \bar{x} = 15 \) and a standard deviation \( s = 3 \), the CV would be \( \left( \frac{3}{15} \right) \times 100\% = 20\% \).
  • This means that the data varies by 20% relative to the mean.
Chebyshev's Theorem and Its Applications
Chebyshev's theorem is a fundamental principle used in statistics to determine the proportion of values that fall within a certain number of standard deviations from the mean, applicable to any dataset regardless of the distribution. This theorem is particularly useful for making estimates without assuming a normal distribution.
According to Chebyshev's theorem:
  • The proportion of values within \( k \) standard deviations of the mean is at least \( 1 - \frac{1}{k^2} \).
  • For example, if you want an interval that contains at least 75% of data values, we solve for \( k \) such that \( 1 - \frac{1}{k^2} = 0.75 \). Solving gives \( k = 2 \), indicating that around at least 75% of data lies within two standard deviations of the mean.
  • With a mean of 15 and a standard deviation of 3, this gives the interval \( 15 \pm 6 = [9, 21] \).
Exploring Standard Deviation
Standard deviation is a widely used measure of the amount of variation or dispersion in a set of values. It's a crucial statistic that helps us understand how spread out the data points in a dataset are from the mean. A lower standard deviation indicates that the data points tend to be close to the mean, while a higher standard deviation indicates that the data is spread out over a broader range of values.
In practice:
  • Standard deviation is denoted by \( s \) in the context of a sample.
  • To calculate the standard deviation of a dataset, the differences between each data point and the mean are squared, averaged, and then the square root is taken.
  • For example, for our exercise, the standard deviation given is 3, which indicates that data points typically lie 3 units away from the mean, on average.
Understanding standard deviation supports the utilization of other concepts like the coefficient of variation and Chebyshev's theorem, as these depend on an accurate reading of data spread.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the mean, median, and mode of the data set \(\begin{array}{lllll}8 & 2 & 7 & 2 & 6\end{array}\)

Consider the mode, median, and mean. Which average represents the middle value of a data distribution? Which average represents the most frequent value of a distribution? Which average takes all the specific values into account?

The Hill of Tara in Ireland is a place of great archaeological importance. This region has been occupied by people for more than 4000 years. Geomagnetic surveys detect subsurface anomalies in the earth's magnetic field. These surveys have led to many significant archaeological discoveries. After collecting data, the next step is to begin a statistical study. The following data measure magnetic susceptibility (centimeter-gram-second \(\times 10^{-6}\) ) on two of the main grids of the Hill of Tara (Reference: Tara: An Archaeological Survey by Conor Newman, Royal Irish Academy, Dublin). Grid E: \(x\) variable \begin{tabular}{lrlllll} \(13.20\) & \(5.60\) & \(19.80\) & \(15.05\) & \(21.40\) & \(17.25\) & \(27.45\) \\ \(16.95\) & \(23.90\) & \(32.40\) & \(40.75\) & \(5.10\) & \(17.75\) & \(28.35\) \\ Grid H: \(y\) variable & & & & & \\ \(11.85\) & \(15.25\) & \(21.30\) & \(17.30\) & \(27.50\) & \(10.35\) & \(14.90\) \\ \(48.70\) & \(25.40\) & \(25.95\) & \(57.60\) & \(34.35\) & \(38.80\) & \(41.00\) \\ \(31.25\) & & & & & & \\ \hline \end{tabular} (a) Compute \(\Sigma x, \Sigma x^{2}, \Sigma y\), and \(\Sigma y^{2}\). (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\) and for \(y\). (c) Compute a \(75 \%\) Chebyshev interval around the mean for \(x\) values and also for \(y\) values. Use the intervals to compare the magnetic susceptibility on the two grids. Higher numbers indicate higher magnetic susceptibility. However, extreme values, high or low, could mean an anomaly and possible archaeological treasure. (d) Compute the sample coefficient of variation for each grid. Use the \(\mathrm{CV}\) s to compare the two grids. If \(s\) represents variability in the sig- -nal (magnetic susceptibility) and \(\bar{x}\) represents the expected level of the signal, then \(s / \bar{x}\) can be thought of as a measure of the variability per unit of expected signal. Remember, a considerable variability in the signal (above or below average) might indicate buried artifacts. Why, in this case, would a large \(\mathrm{CV}\) be better, or at least more exciting? Explain.

Consider the following ordered data: \(\begin{array}{ccccccccc}2 & 5 & 5 & 6 & 7 & 7 & 8 & 9 & 10\end{array}\) (a) Find the low, \(Q_{1}\), median, \(Q_{3}\), high. (b) Find the interquartile range. (c) Make a box-and-whisker plot.

Pax World Balanced is a highly respected, socially responsible mutual fund of stocks and bonds (see Viewpoint). Vanguard Balanced Index is another highly regarded fund that represents the entire U.S. stock and bond market (an index fund). The mean and standard deviation of annualized percent returns are shown below. The annualized mean and standard deviation are based on the years 1993 through 2002 (Source: Morningstar). Pax World Balanced: \(\bar{x}=9.58 \% ; s=14.05 \%\) Vanguard Balanced Index: \(\bar{x}=9.02 \% ; s=12.50 \%\) (a) Compute the coefficient of variation for each fund. If \(\bar{x}\) represents return and \(s\) represents risk, then explain why the coefficient of variation can be taken to represent risk per unit of return. From this point of view, which fund appears to be better? Explain. (b) Compute a \(75 \%\) Chebyshev interval around the mean for each fund. Use the intervals to compare the two funds. As usual, past performance does not guarantee future performance.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.