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Lecturer’s desk

s c r e e n. s c r e e n. Lecturer’s desk. Row A. Row A. 13. 12. 11. 10. 9. 8. 7. 17. 16. 15. 14. Row A. 19. 18. 4. 3. 2. 1. 6. 5. Row B. 14. 13. 12. 11. 10. 9. 15. Row B. 8. 7. 20. 4. 3. 2. 1. 19. 18. 17. 16. 6. 5. Row B. Row C. 4. 3. 2. 1.

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Lecturer’s desk

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  1. s c r e e n s c r e e n Lecturer’s desk Row A Row A 13 12 11 10 9 8 7 17 16 15 14 Row A 19 18 4 3 2 1 6 5 Row B 14 13 12 11 10 9 15 Row B 8 7 20 4 3 2 1 19 18 17 16 6 5 Row B Row C 4 3 2 1 15 14 13 12 11 10 9 19 18 17 16 Row C 8 7 6 5 21 20 Row C Row D 20 19 18 17 22 21 4 3 2 1 16 15 14 13 12 11 10 9 Row D Row D 8 7 6 5 21 20 19 18 Row E 23 22 4 3 2 1 6 5 16 15 14 13 12 11 10 9 Row E Row E 8 7 17 21 20 19 18 Row F 23 22 4 3 2 1 Row F 6 5 17 16 15 14 13 12 11 10 9 Row F 8 7 22 21 20 19 17 16 15 14 13 12 11 10 9 Row G Row G 8 7 24 23 18 4 3 2 1 6 5 Row G 16 20 19 18 17 Row H 22 21 4 3 2 1 15 14 13 12 11 10 9 Row H 8 7 6 5 Row H table Row J Row J 25 24 23 22 1 18 table 9 6 26 5 20 19 21 13 8 7 14 26 25 24 23 4 3 2 1 27 5 20 22 21 14 13 12 11 10 9 6 18 17 16 15 Row K Row K 8 7 19 27 26 25 24 4 3 2 1 19 18 28 5 20 23 22 21 14 13 12 11 10 9 6 15 Row L Row L 8 7 17 16 4 3 2 1 27 26 25 24 22 21 15 Row M Row M 5 28 23 20 19 14 13 12 11 10 9 6 18 17 16 8 7 22 21 29 28 27 26 4 3 2 1 20 23 19 15 14 18 17 16 25 24 30 5 13 12 11 10 9 6 Row N Row N 8 7 29 28 27 26 22 21 30 4 3 2 1 20 23 19 15 14 18 17 16 25 24 5 13 12 11 10 9 6 Row P Row P 8 7 29 28 27 26 39 38 37 36 30 4 3 2 1 32 31 23 22 21 - 15 14 25 24 40 5 33 35 34 13 12 11 10 9 6 Row Q 8 7 Physics- atmospheric Sciences (PAS) - Room 201

  2. Use this as your study guide By the end of lecture today10/8/12 Law of Large Numbers Central Limit Theorem Three propositions True mean 2) Standard Error of Mean 3) Normal Shape Calculating Confidence Intervals http://onlinestatbook.com/stat_sim/sampling_dist/index.html http://www.youtube.com/watch?v=ne6tB2KiZuk

  3. Introduction to Statistics for the Social SciencesSBS200, COMM200, GEOG200, PA200, POL200, or SOC200Lecture Section 001, Fall, 2012Room 201 Physics and Atmospheric Sciences (PAS)10:00 - 10:50 Mondays, Wednesdays & Fridays. Welcome Please double check – Allcell phones other electronic devices are turned off and stowed away http://www.youtube.com/watch?v=oSQJP40PcGI

  4. Schedule of readings Before next exam this Friday (October 12th) Please read chapters 5, 6, & 8 Please read Chapters 10, 11, 12 and 14 in Plous Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability and Risk Chapter 14: The Perception of Randomness Study guide is online

  5. Lab sessions Labs continuethis week

  6. Please click in My last name starts with a letter somewhere between A. A – D B. E – L C. M – R D. S – Z Please hand in your homework

  7. Homework due – Wednesday (October 10th) On class website: Please print and complete homework worksheet #14 Confidence Intervals

  8. Central Limit Theorem

  9. Sampling distributions of sample means versus frequency distributions of individual scores Distribution of raw scores: is an empirical probability distribution of the values from a sample of raw scores from a population Eugene X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X • Frequency distributions of individual scores • derived empirically • we are plotting raw data • this is a single sample Melvin X X X X X X X X X X X X Take a single score x Repeat over and over x x x Population x x x x

  10. Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population important note: “fixed n” • Sampling distributions of sample means • theoretical distribution • we are plotting means of samples Take sample – get mean Repeat over and over Population

  11. Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population important note: “fixed n” • Sampling distributions of sample means • theoretical distribution • we are plotting means of samples Take sample – get mean Repeat over and over Population Distribution of means of samples

  12. Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population Eugene • Frequency distributions of individual scores • derived empirically • we are plotting raw data • this is a single sample X X X X X X X Melvin X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X • Sampling distributions sample means • theoretical distribution • we are plotting means of samples 23rd sample 2nd sample

  13. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Sampling distribution for continuous distributions • Central Limit Theorem: If random samples of a fixed N are drawn • from any population (regardless of the shape of the • population distribution), as N becomes larger, the • distribution of sample means approaches normality, with • the overall mean approaching the theoretical population • mean. Distribution of Raw Scores Sampling Distribution of Sample means Melvin 23rd sample Eugene X X X X X 2nd sample

  14. Sampling distributions of sample means versus frequency distributions of individual scores • Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population Eugene X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X An example of frequency distributions for population of individual scores Melvin X X X X X X X X µ= 100 X X Mean = 100 X X σ= 3 100 Notice: SEM is smaller than SD – especially as n increases Standard Deviation = 3 23rd sample An example of a sampling distribution of sample means 2nd sample µ = 100 Mean = 100 = 1 Standard Error of the Mean = 1 100

  15. Sampling distributions of sample means versus frequency distributions of individual scores • Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population Eugene X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X An example of frequency distributions for population of individual scores Melvin X X X X X X X X µ= 100 X X Mean = 100 X X σ= 3 100 Notice: SEM is smaller than SD – especially as n increases Standard Deviation = 3 23rd sample An example of a sampling distribution of sample means 2nd sample µ = 100 Mean = 100 = 1 Standard Error of the Mean = 1 100

  16. Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population Notice: SEM is smaller than SD – especially as n increases X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X µ= 100 X X Mean = 100 X X σ= 3 100 Standard Deviation = 3 An example of a sampling distribution of sample means µ = 100 Mean = 100 = 1 Standard Error of the Mean = 1 100

  17. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Sampling distribution for continuous distributions • Central Limit Theorem: If random samples of a fixed N are drawn • from any population (regardless of the shape of the • population distribution), as N becomes larger, the • distribution of sample means approaches normality, with • the overall mean approaching the theoretical population • mean. Distribution of Raw Scores Sampling Distribution of Sample means Melvin Eugene X X X X X

  18. Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population Eugene • Frequency distributions of individual scores • derived empirically • we are plotting raw data • this is a single sample X X X X X X X Melvin X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X • Sampling distributions sample means • theoretical distribution • we are plotting means of samples 23rd sample 2nd sample Important

  19. Sampling distributions of sample means versus frequency distributions of individual scores • Sampling distribution: is a theoretical probability distribution of • the possible values of some sample statistic that would • occur if we were to draw an infinite number of same-sized • samples from a population In principle, sampling distributions exist for means, standard deviations, proportions and correlations (among others)

  20. Central Limit Theorem Proposition 1: If sample size (n) is large enough (e.g. 100) The mean of the sampling distribution will approach the mean of the population As n ↑ x will approach µ Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population As n ↑ curve will approach normal shape Proposition 3: The standard deviation of the sampling distribution equals the standard deviation of the population divided by the square root of the sample size. As n increases SEM decreases. As n ↑ curve variability gets smaller X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

  21. Central Limit Theorem Proposition 1: If sample size (n) is large enough (e.g. 100) The mean of the sampling distribution will approach the mean of the population Law of large numbers: As the number of measurements increases the data becomes more stable and a better approximation of the true (theoretical) probability. Larger sample sizes tend to be associated with stability. As the number of observations (n) increases or the number of times the experiment is performed, the estimate will become more accurate.

  22. Take sample (n = 5) – get mean Proposition 2: If sample size (n) is large enough (e.g. 100), the sampling distribution of means will be approximately normal, regardless of the shape of the population Repeat over and over Population population population population sampling distribution n = 2 sampling distribution n = 5 sampling distribution n = 4 sampling distribution n = 30 sampling distribution n = 5 sampling distribution n = 25

  23. Central Limit Theorem Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population

  24. Central Limit Theorem Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population

  25. Central Limit Theorem Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population

  26. Central Limit Theorem Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population • If sample size (n) is not large (but the shape of the • population is normal), the sampling distribution of • means will be normal. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

  27. Central Limit Theorem Proposition 3: The standard deviation of the sampling distribution equals the standard deviation of the population divided by the square root of the sample size. As n increases SEM decreases. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

  28. Central Limit Theorem Proposition 1: If sample size (n) is large enough (e.g. 100) The mean of the sampling distribution will approach the mean of the population As n ↑ x will approach µ Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population As n ↑ curve will approach normal shape Proposition 3: The standard deviation of the sampling distribution equals the standard deviation of the population divided by the square root of the sample size. As n increases SEM decreases. As n ↑ curve variability gets smaller X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

  29. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Sampling distribution of sample means • Central Limit Theorem: If random samples of a fixed N are drawn • from any population (regardless of the shape of the • population distribution), as N becomes larger, the • distribution of sample means approaches normality, with • the overall mean approaching the theoretical population • mean. Distribution of Raw Scores Sampling Distribution of Sample means Melvin Eugene X X X X X

  30. Central Limit Theorem: If random samples of a fixed N are drawn from any population (regardless of the shape of the population distribution), as N becomes larger, the distribution of sample means approaches normality, with the overall mean approaching the theoretical population mean. Distribution of Raw Scores Animation for creating sampling distribution of sample means Distribution of single sample Eugene Melvin Sampling Distribution of Sample means Sampling Distribution of Sample means Mean for sample 12 Mean for sample 7 http://onlinestatbook.com/stat_sim/sampling_dist/index.html

  31. X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Sampling distribution for continuous distributions • Central Limit Theorem: If random samples of a fixed N are drawn • from any population (regardless of the shape of the • population distribution), as N becomes larger, the • distribution of sample means approaches normality, with • the overall mean approaching the theoretical population • mean. Distribution of Raw Scores Sampling Distribution of Sample means Melvin 23rd sample Eugene X X X X X 2nd sample

  32. Central Limit Theorem Proposition 1: If sample size (n) is large enough (e.g. 100) The mean of the sampling distribution will approach the mean of the population As n ↑ x will approach µ Proposition 2: If sample size (n) is large enough (e.g. 100) The sampling distribution of means will be approximately normal, regardless of the shape of the population As n ↑ curve will approach normal shape Proposition 3: The standard deviation of the sampling distribution equals the standard deviation of the population divided by the square root of the sample size. As n increases SEM decreases. As n ↑ curve variability gets smaller X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

  33. Thank you! See you next time!!

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