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Normal Distribution

Normal Distribution. To understand the normal distribution To be able to find probabilities given the Z score To be able to find the Z score given the probability. Most commonly observed probability distribution.

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Normal Distribution

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  1. Normal Distribution To understand the normal distribution To be able to find probabilities given the Z score To be able to find the Z score given the probability

  2. Most commonly observed probability distribution • 1800s, German mathematician and physicist Karl Gauss used it to analyse astronomical data • Sometimes called the Gaussian distribution in science.

  3. Normal Distribution • Occurs naturally(e.g. height, weight,..) • Often called a “bell curve” • Centres around the mean 

  4. Normal Distribution • Spread depends on standard deviation  • Percentage of distribution included depends on number of standard deviations from the mean

  5. Properties of Normal Distribution • Symmetrical • Area under curve = 1

  6. Standard Normal Distribution • Mean (=0 • Standard deviation ()=1

  7. Standard Normal Distribution • Tables are provided to help us to calculate the probability for the standard normal distribution , Z • Z-scores are a means of answering the question ``how many standard deviations away from the mean is this observation?''

  8. Tables give us P(Z<z) Find P(Z<1.25) It is vital that you always sketch a graph P(Z<1.25) = 0.8944

  9. Tables give us P(Z<z) Find P(Z>1.25) It is vital that you always sketch a graph P(Z>1.25) = 1- 0.8944 = 0.1056

  10. It is vital that you always sketch a graph a) Find P(Z < 1.52) b) Find P(Z > 2.60) c) Find P(Z < -0.75) d) Find P(-1.18 < Z < 1.43)

  11. SOLUTIONS a) Find P(Z < 1.52) P(Z < 1.52) = 0.9357

  12. SOLUTIONS b) Find P(Z > 2.60) P(Z > 2.60) = 1 - 0.9053 = 0.0047

  13. SOLUTIONS c) Find P(Z < -0.75) P(Z < -0.75) = P(Z > 0.75) P(Z > 0.75) = 1 – P(Z < 0.75) P(Z > 0.75) = 1 – 0.7734 = 0.2266

  14. SOLUTIONS d) Find P(-1.18 < Z < 1.43) P(Z<1.43) = 0.9236 P(Z>1.18) = 1-0.881 P(Z>1.18) = 0.119 P(-1.18<Z<1.43) = 0.9236 - 0.119 = 0.8046

  15. Reversing the processGiven the probability find the value of a in P(Z<a) P(Z<1.25) = 0.8944 P(Z<-0.25) = 0.4013 If the probability is >0.5 then a is positive If the probability is <0.5 then a is negative

  16. It is vital that you always sketch a graph a) P(Z < a) = 0.7611 b) P(Z > a) = 0.0287 c) P(Z < a) = 0.0170 • P(Z > a) = 0.01 • ASK ABOUT THIS ONE

  17. SOLUTIONS a) P(Z < a) = 0.7611 0.7611 a = 0.71

  18. SOLUTIONS b) P(Z > a) = 0.0287 0.9713 0.0287 a = 1.9

  19. SOLUTIONS c) P(Z < a) = 0.0170 0.0170 < 0.5 so a is negative 0.9830 0.0170 z = 2.12 so a = -2.12

  20. SOLUTIONS • P(Z > a) = 0.01 Use percentage points of normal distribution table which gives P(Z>z) a = 2.3263

  21. Normal distribution calculator

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