Math Problem Statement

We are given a normally distributed random variable X with µ = 75 and σ = 5.

Find P (X > 85). Find k such as P(X<k) = 0.997. Interpret your result. Find the two values where the middle 68.27% of the distribution of X lie.

Solution

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Math Problem Analysis

Mathematical Concepts

Probability
Normal Distribution
Standard Normal Distribution
Percentiles

Formulas

Z = (X - µ) / σ
P(Z > value) from standard normal distribution table
k = µ + Z * σ for finding X given Z-value

Theorems

Empirical Rule (68-95-99.7 Rule)
Standard Normal Distribution

Suitable Grade Level

Grades 10-12