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Magazine Chapter 3: Problem 27
Consider a sample with a mean of 30 and a standard deviation of \(5 .\) Use Chebyshev's theorem to determine the percentage of the data within each of the following ranges: a. 20 to 40 b. 15 to 45 c. 22 to 38 d. 18 to 42 e. 12 to 48
Short Answer
Expert verified
Percentages are 75%, 88.89%, 60.94%, 82.64%, and 92.27% for each range.
Step by step solution
01
Understanding Chebyshev's Theorem
Chebyshev's theorem states that for any dataset, regardless of distribution, at least \((1 - \frac{1}{k^2})\times 100\%\) of the data lies within \(k\) standard deviations from the mean. Here, \(k\) is the number of standard deviations from the mean.
02
Calculating k for Range 20 to 40
The mean is 30. The range 20 to 40 spans 10 units below and above the mean. This means: \[ k = \frac{40-30}{5} = 2 \] and \[ k = \frac{30-20}{5} = 2 \].
03
Calculate Percentage for 20 to 40 Using Chebyshev's Theorem
Using the formula from Chebyshev's theorem: \[ \left(1 - \frac{1}{2^2}\right)\times 100\% = 75\% \] Therefore, at least 75% of the data is within 20 to 40.
04
Calculating k for Range 15 to 45
The mean is 30. The range 15 to 45 spans 15 units below and above the mean. This means: \[ k = \frac{45-30}{5} = 3 \] and \[ k = \frac{30-15}{5} = 3 \].
05
Calculate Percentage for 15 to 45 Using Chebyshev's Theorem
Using the formula: \[ \left(1 - \frac{1}{3^2}\right)\times 100\% = 88.89\% \] Therefore, at least 88.89% of the data is within 15 to 45.
06
Calculating k for Range 22 to 38
The mean is 30. The range 22 to 38 spans 8 units below and above the mean. This means: \[ k = \frac{38-30}{5} = 1.6 \] and \[ k = \frac{30-22}{5} = 1.6 \].
07
Calculate Percentage for 22 to 38 Using Chebyshev's Theorem
Using the formula: \[ \left(1 - \frac{1}{1.6^2}\right)\times 100\% \approx 60.94\% \] Therefore, at least 60.94% of the data is within 22 to 38.
08
Calculating k for Range 18 to 42
The mean is 30. The range 18 to 42 spans 12 units below and above the mean. This means: \[ k = \frac{42-30}{5} = 2.4 \] and \[ k = \frac{30-18}{5} = 2.4 \].
09
Calculate Percentage for 18 to 42 Using Chebyshev's Theorem
Using the formula: \[ \left(1 - \frac{1}{2.4^2}\right)\times 100\% \approx 82.64\% \] Therefore, at least 82.64% of the data is within 18 to 42.
10
Calculating k for Range 12 to 48
The mean is 30. The range 12 to 48 spans 18 units below and above the mean. This means: \[ k = \frac{48-30}{5} = 3.6 \] and \[ k = \frac{30-12}{5} = 3.6 \].
11
Calculate Percentage for 12 to 48 Using Chebyshev's Theorem
Using the formula: \[ \left(1 - \frac{1}{3.6^2}\right)\times 100\% \approx 92.27\% \] Therefore, at least 92.27% of the data is within 12 to 48.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Standard Deviation
The standard deviation is a measure that tells us how much variation or spread there is in a set of data. It shows us how much individual data points differ from the mean of the dataset. A smaller standard deviation means the data points are close to the mean, while a larger standard deviation indicates a wider spread.
In our exercise, we have a standard deviation of 5. Here's why it's important:
In our exercise, we have a standard deviation of 5. Here's why it's important:
- It tells us that most data points lie within a certain range around the mean.
- It's crucial for calculating the value of `k` in Chebyshev's theorem, as we divide the deviation of each boundary from the mean by this value.
Chebyshev's Theorem and Percentage of Data
Chebyshev's theorem is a useful statistical tool for understanding how much data lies within given ranges. It states that for any dataset, regardless of its distribution shape, at least \(1 - \frac{1}{k^2}\)脳100% of data falls within `k` standard deviations from the mean.
This principle helps us determine the percentage of data that falls within specific intervals in our exercise. For instance:
This principle helps us determine the percentage of data that falls within specific intervals in our exercise. For instance:
- When `k=2`, at least 75% of the data is within two standard deviations from the mean.
- Similarly, if `k=3`, at least 88.89% of the data is included in this range.
Calculating the Mean
The mean, often known as the average, is a fundamental statistic that sums up a dataset's central tendency. It is calculated by adding up all the data points and then dividing by the number of points. In our specific exercise, the mean is given as 30, central to our calculations.
Here鈥檚 why the mean is essential:
Here鈥檚 why the mean is essential:
- It acts as the center of the data, providing a reference point around which standard deviations are calculated.
- In Chebyshev's theorem, the term \( k \) involves measuring how far from the mean each range extends, and this is critical for calculating the percentage of data.
Role of Sample in Statistics
A sample is part of a larger population and is used to make inferences about the population's behavior. When working with Chebyshev's theorem and other statistical concepts, we frequently work with samples because it鈥檚 often impractical to analyze entire populations.
In our exercise, the sample has given context for calculations like the mean and standard deviation. Some benefits of working with samples include:
In our exercise, the sample has given context for calculations like the mean and standard deviation. Some benefits of working with samples include:
- They provide a manageable way to obtain valuable data insights without the need for exhaustive data processing.
- Allows for predictions and estimates that apply to a larger population context.
