Math Problem Statement

A data set has mean μ and standard deviation σ. Which statement is correct?

Solution

The correct answer is:

"A value chosen randomly from the data set will typically be contained in the interval [μ - σ, μ + σ]."

Explanation:

  • For a dataset with mean μ and standard deviation σ, about 68% of the data values typically fall within one standard deviation from the mean, i.e., in the interval [μ - σ, μ + σ].
  • This follows from the empirical rule (68-95-99.7 rule) for normal distributions, which describes how data is distributed in terms of standard deviations from the mean.

Let me know if you have questions or need further clarification!


Here are 5 questions to deepen your understanding:

  1. What does the standard deviation represent in a data set?
  2. How would the interval change for 95% of the data instead of 68%?
  3. How can this interval differ for non-normally distributed data?
  4. Can a value outside the interval [μ - σ, μ + σ] occur? If so, how often?
  5. What happens to the interval [μ - σ, μ + σ] when the standard deviation is very large?

Tip: Always check whether the distribution is normal when applying the empirical rule!

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Math Problem Analysis

Mathematical Concepts

Statistics
Empirical Rule
Standard Deviation

Formulas

Interval [μ - σ, μ + σ] covers approximately 68% of data for a normal distribution

Theorems

Empirical Rule (68-95-99.7 Rule)

Suitable Grade Level

Grades 9-12