Math Problem Statement
Based on what you have studied in this unit, reflect and answer the following questions:
What are the main characteristics of the normal distribution? Why is it advantageous to use the Z scores?
For a standard normal distribution, about 68% of values falls within one standard deviation of the mean; about 95% falls within two standard deviations; and about 99.7% falls within three standard deviations. Suppose the distribution of math scores in the SAT follow a normal distribution with a mean µ=700 and a standard deviation σ=150.
Calculate the Z-score for an SAT Math score of 600. Interpret it in your own words. Find the interval [a, b] that captures 68% of the SAT Math score in this example. Find the interval [a, b] that captures 95% of the SAT Math score in this example. Find the interval [a, b] that captures 99.7% of the SAT Math score in this example.
Solution
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Math Problem Analysis
Mathematical Concepts
Normal Distribution
Z-Scores
Empirical Rule
Standard Deviations
Formulas
Z = (X - µ) / σ
Interval for 68%: [µ - σ, µ + σ]
Interval for 95%: [µ - 2σ, µ + 2σ]
Interval for 99.7%: [µ - 3σ, µ + 3σ]
Theorems
Empirical Rule (68-95-99.7 rule)
Suitable Grade Level
Grades 10-12
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