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Special Distributions

Special Distributions. The Normal Probability Distribution - a special kind of continuous probability distribution. The Normal Distribution. f(x) -    x. The Normal Distribution. The Mathematical Model. f ( X ) = frequency of random variable X

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Special Distributions

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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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