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Understanding the Normal Probability Distribution: Key Concepts and Calculations

The Normal Probability Distribution is a fundamental concept in statistics, representing continuous random variables that are symmetrically distributed around the mean. This guide covers the mathematical model, including equations for calculating probabilities and understanding standard deviation and mean. It provides examples of using integral calculus and standard normal tables to find probabilities, percentiles, and the interquartile range. Through practical examples, we illustrate how to transform random variables and interpret the results effectively.

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Understanding the Normal Probability Distribution: Key Concepts and Calculations

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  1. Special Distributions The Normal Probability Distribution - a special kind of continuous probability distribution

  2. The Normal Distribution f(x) -   x

  3. The Normal Distribution

  4. The Mathematical Model f(X) = frequency of random variable X  = 3.14159; e = 2.71828  = population standard deviation X = value of random variable (- < X < )  = population mean

  5. Finding Normal Probabilities Via Integral calculus: Given X ~ N (, ), find P(a  X  b) =  f(x) dx  a b

  6. Finding Normal Probabilities Via Standard Normal Probability Tables: - convert the unstandardized normal random variable, X, to a standardized normal random variable, Z, via the transformation: Z = X -   where Z ~ N (, )

  7. Example: X ~ N ( = 410,  = 60) 1. Find P(X  300) 300 =410

  8. Example: P(X  300) = P[(X - )/  (300 – 410)/60] = P[Z  -1.83] -1.83 =0

  9. The Standard Normal Table Z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 0.1 0.2 0.3 . . . 1.8 .4664

  10. Example: P(X  300) = P[(X - )/  (300 – 410)/60] = P[Z  -1.83] = .0336 .5-.4664=.0336 .4664 -1.83 =0

  11. Example: X ~ N ( = 410,  = 60) 2. Find P(350  X  450) 350 =410 450

  12. Example: P(350  X  450) = P[(350 - 410)/60 (X - )/  (450 – 410)/60] = P[-1.00  Z  .67] = .3413 + .2485 = .5899 .3413 .2486 -1.00 =0 .67

  13. Example: X ~ N ( = 410,  = 60) 3. Find the 95th percentile 95%5% =410 xo

  14. Example: The 95th percentile in z-score form: 95%5% =0 zo

  15. Example: The 95th percentile in z-score form: 45% = .4500 95%5% =0 zo

  16. The Standard Normal Table Z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 0.0 0.1 0.2 0.3 . . . 1.6 .4495 .4505

  17. Example: The 95th percentile in z-score form: 45% = .4500 95%5% =0 zo = 1.65

  18. Example: The 95th percentile in X-score form: X =  + Z = 410 + 60(1.65) = 509

  19. Example: X ~ N ( = 410,  = 60) Find the Interquartile Range

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