2. Stage Screen Row B 13 121110 20191817 14 13 121110 19181716 1514 Gallagher Theater 16 65879 Row R 6 58 7 9 Lecturer’s desk Row A Row B Row C 4 3 2 43 2 1 1 43 2 43 2 1 1 43 2 43 2 1 1 43 2 43 2 1 1 43 2 43 2 1 1 43 2 43 2 1 1 43 2 43 2 1 1 43 2 43 2 1 1 3 21 3 2 43 21 Row A 17 16 15 Row A Row C 131211 10 1514 6 58 7 9 Row D 13121110 1514 16 6 58 7 9 20191817 Row D Row E 131211 10 1514 6 58 7 9 19181716 Row E Row F 13121110 1514 16 6 58 7 9 20191817 Row F Row G 13121110 1514 6 58 7 9 19181716 Row G Row H 13121110 1514 16 6 58 7 9 20191817 Row H Row I 13121110 1514 6 58 7 9 19181716 Row I Row J 13121110 1514 16 6 58 7 9 20191817 Row J Row K 13121110 1514 6 58 7 9 19181716 Row K Row L 13121110 1514 16 6 58 7 9 20 191817 Row L Row M 13121110 1514 6 58 7 9 19181716 Row M Row N 13121110 1514 16 6 5879 20191817 Row N Row O 13121110 1514 6 58 7 9 19181716 Row O Row P 13121110 1514 16 6 5879 20191817 Row P Row Q 13121110 6 5879 161514 Row Q 4 4 Row R 10 879 Row S Row B Row C Row D Row E Row F Row G Row H Row I Row J Row K Row L Row M Row N Row O Row P Row Q 26Left-Handed Desks A14, B16, B20, C19, D16, D20, E15, E19, F16, F20, G19, H16, H20, I15, J16, J20, K19, L16, L20, M15, M19, N16, P20, Q13, Q16, S4 5 Broken Desks B9, E12, G9, H3, M17 Need Labels B5, E1, I16, J17, K8, M4, O1, P16 Left handed

3. Stage Screen 2213 121110 Row A Row B Row C Row D Row E Row F Row G Row H Row J Row K Row L Row M 17 Row C Row D Row E Projection Booth 65 4 table Row C Row D Row E 30 27 26252423 282726 2524 23 3127262524 23 R/L handed broken desk 16 1514 13 12 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 Social Sciences 100 Row N Row O Row P Row Q Row R 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 2213 121110 2019181716151421 8 7 9 65 4 8 7 9 3 2 6 5 48793 2 1 6 5 48793 2 1 Row F Row G Row H Row J Row K Row L Row M Row N Row O Row P Row Q Row R 6 5 48793 2 1 6 5 48793 2 1 Row I 2213 121110 2019181716151421 Row I 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 6 5 48793 2 1 Lecturer’s desk 6 5 48793 2 1 262524 23 302928 Row F Row G Row H Row J Row K Row L Row M Row N Row O Row P Row Q Row R Row I 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 3127262524 23 302928 Row B 2928 27

4. MGMT 276: Statistical Inference in Management Fall, 2014 Green sheets

5. Reminder Talking or whispering to your neighbor can be a problem for us – please consider writing short notes. A note on doodling

7. Before our next exam (October 21 st ) Lind (5 – 11) Chapter 5: Survey of Probability Concepts Chapter 6: Discrete Probability Distributions Chapter 7: Continuous Probability Distributions Chapter 8: Sampling Methods and CLT Chapter 9: Estimation and Confidence Interval Chapter 10: One sample Tests of Hypothesis Chapter 11: Two sample Tests of Hypothesis Plous (10, 11, 12 & 14) Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability and Risk Chapter 14: The Perception of Randomness Schedule of readings Study Guide is on the class website

8. On class website: Please print and complete homework worksheet #11 Calculating Confidence Intervals And Examples of Type I and Type II Errors Due October 16 th Extra Credit Opportunity - Also due October 16 th Dan Gilbert Reading and Law of Large Numbers Homework

9. By the end of lecture today 10/14/14 Use this as your study guide Law of Large Numbers Central Limit Theorem

10. Review of Homework Worksheet just in case of questions

11. Homework review Based on apriori probability – all options equally likely – not based on previous experience or data Based on expert opinion - don’t have previous data for these two companies merging together 2 5 =.40 Based on frequency data (Percent of rockets that successfully launched)

12. Homework review Based on apriori probability – all options equally likely – not based on previous experience or data Based on frequency data (Percent of times that pages that are “fake”) 30 100 =.30 Based on frequency data (Percent of times at bat that successfully resulted in hits)

13. Homework review 5 50 =.10 Based on frequency data (Percent of students who successfully chose to be Economics majors)

14. . 44 - 50 4 = -1.5 55 - 50 4 = +1.25 z of 1.5 = area of.4332.4332 +.3944 =.8276 z of 1.25 = area of.3944 55 - 50 4 = +1.25.5000 -.3944 =.1056 1.25 = area of.3944.3944 52 - 50 4 = +.5 55 - 50 4 = +1.25 z of.5 = area of.1915.3944 -.1915 =.2029 z of 1.25 = area of.3944.3944.1915.8276.1056.2029.4332.3944

15. Homework review 3000 - 2708 650 = 0.45 z of 0.45 = area of.1736.5000 -.1736 =.3264.1736 3000 - 2708 650 = 0.45 z of 0.45 = area of.1736.3888 -.1736 =.2152 3500 - 2708 650 = 1.22 z of 1.22 = area of.3888.1736.1255 2500 - 2708 650 = -.32 z of -0.32 = area of.1255.3888 +.1255=.5143 3500 - 2708 650 = 1.22 z of 1.22 = area of.3888.3888.3264.2152.5143.3888

16. Homework review 20 - 15 3.5 = 1.43 z of 1.43 = area of.4236.5000 -.4236 =.0764.4236 20 - 15 3.5 = 1.43 z of 1.43 = area of.4236 z of -1.43 = area of.4236.4236 –.3051 =.1185 z of -.86 = area of.3051.4236.5000 +.4236 =.9236 10 - 15 3.5 = -1.43 12 - 15 3.5 = -0.86.0764.1185.9236.3051.4236

17. Comments on Dan Gilbert Reading

18. Law of large numbers: As the number of measurements increases the data becomes more stable and a better approximation of the true (theoretical) probability As the number of observations (n) increases or the number of times the experiment is performed, the estimate will become more accurate.

19. Law of large numbers: As the number of measurements increases the data becomes more stable and a better approximation of the true signal (e.g. mean) As the number of observations (n) increases or the number of times the experiment is performed, the signal will become more clear (static cancels out) http://www.youtube.com/watch?v=ne6tB2KiZuk With only a few people any little error is noticed (becomes exaggerated when we look at whole group) With many people any little error is corrected (becomes minimized when we look at whole group)

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

21. 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 Sampling distributions of sample means theoretical distribution we are plotting means of samples Population Take sample – get mean Repeat over and over important note: “fixed n” Mean for 1 st sample

22. 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 Population Distribution of means of samples Sampling distributions of sample means theoretical distribution we are plotting means of samples Take sample – get mean Repeat over and over important note: “fixed n”

23. 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 XXXXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXX XXXXXX XXXX XXXX X X XXXXXXXXXX XXXXXXXXXX X XXXXXXXXXX 2 nd sample 23 rd sample Sampling distributions sample means theoretical distribution we are plotting means of samples Frequency distributions of individual scores derived empirically we are plotting raw data this is a single sample Melvin Eugene

24. 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. Sampling distribution for continuous distributions XXXXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXX XXXXXX XXXX XXXX X X XXXXXXXXXX XXXXXXXXXX X XXXXXXXXXX Melvin Eugene Sampling Distribution of Sample means Distribution of Raw Scores 2 nd sample 23 rd sample

25. An example of a sampling distribution of sample means µ = 100 σ = 3 = 1 XXXXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXX XXXXXX XXXX XXXX X X XXXXXXXXXX XXXXXXXXXX X XXXXXXXXXX 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 Mean = 100 100 Standard Deviation = 3 µ = 100 Mean = 100 Standard Error of the Mean = 1 Notice: SEM is smaller than SD – especially as n increases Melvin Eugene 2 nd sample 23 rd sample

26. 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 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 XXXXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXX XXXXXX XXXX XXXX X X XXXXXXXXXX XXXXXXXXXX X XXXXXXXXXX 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 ↑ x will approach µ As n ↑ curve will approach normal shape As n ↑ curve variability gets smaller

27. 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 Central Limit Theorem 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.

28. 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 population sampling distribution n = 5 sampling distribution n = 30 sampling distribution n = 2 sampling distribution n = 5 sampling distribution n = 4 sampling distribution n = 25 Population Take sample (n = 5) – get mean Repeat over and over

29. 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

30. 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

31. 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

32. Central Limit Theorem XXXXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXX XXXXXX XXXX XXXX X X XXXXXXXXXX XXXXXXXXXX X XXXXXXXXXX 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.

33. 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 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 XXXXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXX XXXXXX XXXX XXXX X X XXXXXXXXXX XXXXXXXXXX X XXXXXXXXXX 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 ↑ x will approach µ As n ↑ curve will approach normal shape As n ↑ curve variability gets smaller

34. Central Limit Theorem XXXXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXX XXXXXX XXXX XXXX X X XXXXXXXXXX XXXXXXXXXX X XXXXXXXXXX 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.

35. 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 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 XXXXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXX XXXXXX XXXX XXXX X X XXXXXXXXXX XXXXXXXXXX X XXXXXXXXXX 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 ↑ x will approach µ As n ↑ curve will approach normal shape As n ↑ curve variability gets smaller

36. 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. Animation for creating sampling distribution of sample means http://onlinestatbook.com/stat_sim/sampling_dist/index.html Eugene Melvin Mean for sample 12 Mean for sample 7 Distribution of Raw Scores Sampling Distribution of Sample means Distribution of single sample Sampling Distribution of Sample means

37. 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. Sampling distribution for continuous distributions XXXXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXX XXXXXX XXXX XXXX X X XXXXXXXXXX XXXXXXXXXX X XXXXXXXXXX Melvin Eugene Sampling Distribution of Sample means Distribution of Raw Scores 2 nd sample 23 rd sample

38. 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 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 XXXXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXX XXXXXX XXXX XXXX X X XXXXXXXXXX XXXXXXXXXX X XXXXXXXXXX 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 ↑ x will approach µ As n ↑ curve will approach normal shape As n ↑ curve variability gets smaller

39. . Writing Assignment: Writing a letter to a friend Imagine you have a good friend (pick one). This is a good friend whom you consider to be smart and interested in stuff generally. They are teaching themselves stats (hoping to test out of the class) but need your help on a couple ideas. For this assignment please write your friend/mom/dad/ favorite cousin a letter answering these five questions: (Feel free to use diagrams and drawings if you think that can help) Dear Friend, 1. I’m struggling with this whole Central Limit Theorem idea. Could you describe for me the difference between a distribution of raw scores, and a distribution of sample means? 2. I also don’t get the “three propositions of the Central Limit Theorem”. They all seem to address sample size, but I don’t get how sample size could affect these three things. If you could help explain it, that would be really helpful.

40. . Imagine you have a good friend (pick one). This is a good friend whom you consider to be smart and interested in stuff generally. They are teaching themselves stats (hoping to test out of the class) but need your help on a couple ideas. For this assignment please write your friend/mom/dad/ favorite cousin a letter answering these five questions: (Feel free to use diagrams and drawings if you think that can help) Dear Friend, 1. I’m struggling with this whole Central Limit Theorem idea. Could you describe for me the difference between a distribution of raw scores, and a distribution of sample means? 2. I also don’t get the “three propositions of the Central Limit Theorem”. They all seem to address sample size, but I don’t get how sample size could affect these three things. If you could help explain it, that would be really helpful.