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  1. Last Time • Q-Q plots • Q-Q Envelope to understand variation • Applications of Normal Distribution • Population Modeling • Measurement Error • Law of Averages • Part 1: Square root law for s.d. of average • Part 2: Central Limit Theorem Averages tend to normal distribution

  2. Reading In Textbook Approximate Reading for Today’s Material: Pages 61-62, 66-70, 59-61, 335-346 Approximate Reading for Next Class: Pages 322-326, 337-344, 488-498

  3. Applications of Normal Dist’n • Population Modeling Often want to make statements about: • The population mean, μ • The population standard deviation, σ Based on a sample from the population Which often follows a Normal distribution Interesting Question: How accurate are estimates? (will develop methods for this)

  4. Applications of Normal Dist’n • Measurement Error Model measurement X = μ + e When additional accuracy is required, can make several measurements, and average to enhance accuracy Interesting question: how accurate? (depends on number of observations, …) (will study carefully & quantitatively)

  5. Random Sampling Useful model in both settings 1 & 2: Set of random variables Assume: a. Independent b. Same distribution Say: are a “random sample”

  6. Law of Averages Law of Averages, Part 1: Averaging increases accuracy, by factor of

  7. Law of Averages Recall Case 1: CAN SHOW: Law of Averages, Part 2 So can compute probabilities, etc. using: • NORMDIST • NORMINV

  8. Law of Averages Case 2: any random sample CAN SHOW, for n “large” is “roughly” Consequences: • Prob. Histogram roughly mound shaped • Approx. probs using Normal • Calculate probs, etc. using: • NORMDIST • NORMINV

  9. Law of Averages Case 2: any random sample CAN SHOW, for n “large” is “roughly” Terminology: • “Law of Averages, Part 2” • “Central Limit Theorem” (widely used name)

  10. Central Limit Theorem For any random sample and for n “large” is “roughly”

  11. Central Limit Theorem For any random sample and for n “large” is “roughly” Some nice illustrations

  12. Central Limit Theorem For any random sample and for n “large” is “roughly” Some nice illustrations: • Applet by Webster West & Todd Ogden

  13. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll single die

  14. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll single die For 10 plays

  15. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll single die For 10 plays, histogram

  16. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll single die For 20 plays, histogram

  17. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll single die For 50 plays, histogram

  18. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll single die For 100 plays, histogram

  19. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll single die For 1000 plays, histogram

  20. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll single die For 10,000 plays, histogram

  21. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll single die For 100,000 plays, histogram Stabilizes at Uniform

  22. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll 5 dice For 1 play, histogram

  23. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll 5 dice For 10 plays, histogram

  24. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll 5 dice For 100 plays, histogram

  25. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll 5 dice For 1000 plays, histogram

  26. Central Limit Theorem Illustration: West – Ogden Applet http://www.amstat.org/publications/jse/v6n3/applets/CLT.html Roll 5 dice For 10,000 plays, histogram Looks mound shaped

  27. Central Limit Theorem For any random sample and for n “large” is “roughly” Some nice illustrations: • Applet by Webster West & Todd Ogden • Applet from Rice Univ.

  28. Central Limit Theorem Illustration: Rice Univ. Applet http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html Starting Distribut’n

  29. Central Limit Theorem Illustration: Rice Univ. Applet http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html Starting Distribut’n user input

  30. Central Limit Theorem Illustration: Rice Univ. Applet http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html Starting Distribut’n user input (very non-Normal)

  31. Central Limit Theorem Illustration: Rice Univ. Applet http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html Starting Distribut’n user input (very non-Normal) Dist’n of average of n = 2

  32. Central Limit Theorem Illustration: Rice Univ. Applet http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html Starting Distribut’n user input (very non-Normal) Dist’n of average of n = 2 (slightly more mound shaped?)

  33. Central Limit Theorem Illustration: Rice Univ. Applet http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html Starting Distribut’n user input (very non-Normal) Dist’n of average of n = 5 (little more mound shaped?)

  34. Central Limit Theorem Illustration: Rice Univ. Applet http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html Starting Distribut’n user input (very non-Normal) Dist’n of average of n = 10 (much more mound shaped?)

  35. Central Limit Theorem Illustration: Rice Univ. Applet http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html Starting Distribut’n user input (very non-Normal) Dist’n of average of n = 25 (seems very mound shaped?)

  36. Central Limit Theorem For any random sample and for n “large” is “roughly” Some nice illustrations: • Applet by Webster West & Todd Ogden • Applet from Rice Univ. • Stats Portal Applet

  37. Central Limit Theorem Illustration: StatsPortal. Applet http://courses.bfwpub.com/ips6e Density for average of n = 1, from Exponential dist’n

  38. Central Limit Theorem Illustration: StatsPortal. Applet http://courses.bfwpub.com/ips6e Density for average of n = 1, from Exponential dist’n Best fit Normal density

  39. Central Limit Theorem Illustration: StatsPortal. Applet http://courses.bfwpub.com/ips6e Density for average of n = 2, from Exponential dist’n Best fit Normal density

  40. Central Limit Theorem Illustration: StatsPortal. Applet http://courses.bfwpub.com/ips6e Density for average of n = 4, from Exponential dist’n Best fit Normal density

  41. Central Limit Theorem Illustration: StatsPortal. Applet http://courses.bfwpub.com/ips6e Density for average of n = 10, from Exponential dist’n Best fit Normal density

  42. Central Limit Theorem Illustration: StatsPortal. Applet http://courses.bfwpub.com/ips6e Density for average of n = 30, from Exponential dist’n Best fit Normal density

  43. Central Limit Theorem Illustration: StatsPortal. Applet http://courses.bfwpub.com/ips6e Density for average of n = 100, from Exponential dist’n Best fit Normal density

  44. Central Limit Theorem Illustration: StatsPortal. Applet http://courses.bfwpub.com/ips6e Very strong Convergence For n = 100

  45. Central Limit Theorem Illustration: StatsPortal. Applet http://courses.bfwpub.com/ips6e Looks “pretty good” For n = 30

  46. Central Limit Theorem For any random sample and for n “large” is “roughly” How large n is needed?

  47. Central Limit Theorem How large n is needed?

  48. Central Limit Theorem How large n is needed? • Depends completely on setup

  49. Central Limit Theorem How large n is needed? • Depends completely on setup • Indiv. obs. Close to normal  small OK

  50. Central Limit Theorem How large n is needed? • Depends completely on setup • Indiv. obs. Close to normal  small OK • But can be large in extreme cases