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Chapter 3: Problem 74

A sample of 2000 observations has a mean of 74 and a standard deviation of 12 . Using Chebyshev's theorem, find at least what percentage of the observations fall in the intervals \(\bar{x} \pm 2 s, \bar{x} \pm 2.5 s\), and \(\bar{x} \pm 3 s\). Note that here \(\bar{x} \pm 2 s\) represents the interval \(\bar{x}-2 s\) to \(\bar{x}+2 s\), and so on.

Short Answer

Expert verified
At least 75% of the observations fall within the interval 74卤24, at least 84% of the observations fall within the interval 74卤30, and at least 89% of the observations fall within the interval 74卤36.

Step by step solution

01

Identify values

Directly from the exercise, we have the mean \(\bar{x} = 74\), the standard deviation \(s = 12\), and the values of k given as \(k = 2\), \(k = 2.5\), and \(k = 3\).
02

Apply Chebyshev's theorem for k = 2

Putting \(k = 2\) in Chebyshev's theorem, we get \(1 - (1/k^2) = 1 - 1/4 = 0.75\) or 75% of the samples fall within an interval of \(\bar{x} \pm 2s = 74 \pm 2*12\).
03

Apply Chebyshev's theorem for k = 2.5

Putting \(k = 2.5\) in Chebyshev's theorem, we get \(1 - (1/k^2) = 1 - 1/6.25 = 0.84\) or 84% of the samples fall within an interval of \(\bar{x} \pm 2.5s = 74 \pm 2.5*12\).
04

Apply Chebyshev's theorem for k = 3

Putting \(k = 3\) in Chebyshev's theorem, we get \(1 - (1/k^2) = 1 - 1/9 = 0.89\) or 89% of the samples fall within an interval of \(\bar{x} \pm 3s = 74 \pm 3*12\).

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Key Concepts

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

Understanding the Mean
The mean is a central concept in statistics often referred to as the "average." It is calculated by summing all the observations and dividing by the total number of observations. In our example, a sample of 2000 observations has a mean of 74. This mean provides a central value around which data points are distributed.

Think of the mean as a balancing point. It gives us an idea of where most values cluster. When analyzing data, knowing the mean helps us further explore how other values vary. This is where measures like standard deviation come into play.
Demystifying Standard Deviation
Standard deviation measures how spread out the numbers in a dataset are. A low standard deviation means the numbers are close to the mean, while a high standard deviation indicates a wide range of values. In this context, the standard deviation is 12.

With a mean of 74 and a standard deviation of 12,
  • Numbers close to 74 are common.
  • Values much higher or lower than 74 are further from the average.
Understanding this spread helps us apply Chebyshev's theorem, which uses standard deviation to estimate how many observations fall within a certain distance from the mean.
Interval Estimation with Chebyshev's Theorem
Interval estimation allows us to understand the proportion of data within specific ranges. Using Chebyshev's theorem, we can determine what percentage of observations lie within certain standard deviations from the mean.

Chebyshev's theorem states that for any dataset, at least \(1 - \frac{1}{k^2}\) of the observations fall within \(k\) standard deviations from the mean:
  • When \(k = 2\), at least 75% of observations fall between \(74 \pm 24\).
  • When \(k = 2.5\), 84% fall between \(74 \pm 30\).
  • When \(k = 3\), 89% are within \(74 \pm 36\).
This principle is powerful because it applies to any shaped distribution, providing a way to gauge data dispersion without knowing the precise distribution type.

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Most popular questions from this chapter

Which of the three measures of central tendency (the mean, the median, and the mode) can be calculated for quantitative data only, and which can be calculated for both quantitative and qualitative data? Illustrate with examples.

The mean time taken to learn the basics of a word processor by all students is 200 minutes with a standard deviation of 20 minutes. a. Using Chebyshev's theorem, find at least what percentage of students will learn the basics of this word processor in i. 160 to 240 minutes ii. 140 to 260 minutes \({ }^{*}\) b. Using Chebyshev's theorem, find the interval that contains the time taken by at least \(75 \%\) of all students to learn this word processor.

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 mean monthly mortgage paid by all home owners in a town is \(\$ 2365\) with a standard deviation of \(\$ 340\) a. Using Chebyshev's theorem, find at least what percentage of all home owners in this town pay a monthly mortgage of i. \(\$ 1685\) to \(\$ 3045\) ii. \(\$ 1345\) to \(\$ 3385\) \({ }^{*} \mathbf{b}\). Using Chebyshev's theorem, find the interval that contains the monthly mortgage payments of at least \(84 \%\) of all home owners.

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 2000 of the Metro Business College who were voted most likely to succeed. \(\begin{array}{llllllllll}59 & 68 & 84 & 78 & 107 & 382 & 56 & 74 & 97 & 60\end{array}\) 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.
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