Math Problem Statement
Solution
The image you uploaded presents a data set of exam scores, along with the sample mean of 83.5 and a sample standard deviation of 16. The question is asking why Chebyshev's Rule was applied for this data set.
Chebyshev’s Rule Explanation:
Chebyshev's Rule, or Chebyshev's Theorem, applies to any data distribution, regardless of its shape. This is a key difference between Chebyshev's Rule and the Empirical Rule, which only applies to normal (bell-shaped) distributions. Chebyshev’s Theorem states that for any data set, at least of the data lies within standard deviations of the mean, where .
Answer Explanation:
- The reason for using Chebyshev's Rule here is likely because the data set is not symmetric. In cases where the data does not follow a normal distribution (symmetry), the Empirical Rule (which assumes normality) is not appropriate. Chebyshev’s Rule is used because it applies universally to all data sets, symmetric or not.
Conclusion:
The correct answer would be "The data set was not symmetric."
Would you like more details on this or clarification?
5 Related Questions:
- What are the conditions for applying the Empirical Rule versus Chebyshev’s Rule?
- How does the shape of a data distribution affect the choice of statistical rules?
- Can Chebyshev's Rule give more precise estimates for symmetric data distributions?
- What percentage of data lies within 2 standard deviations according to Chebyshev’s Rule?
- How do you calculate the standard deviation of a sample manually?
Tip:
If your data appears skewed or non-normal, always consider using Chebyshev's Rule instead of the Empirical Rule for making predictions about the data distribution.
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Math Problem Analysis
Mathematical Concepts
Statistics
Data Distribution
Standard Deviation
Chebyshev's Rule
Formulas
Chebyshev's Rule: 1 - (1/k^2) where k > 1
Theorems
Chebyshev's Theorem
Suitable Grade Level
Grades 9-12
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