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

MGMT 276: Statistical Inference in Management Fall 2015 Welcome

Just for Fun Assignments Go to D2L - Click on “Content” Click on “Interactive Online Just-for-fun Assignments” Please note: These are not worth any class points and are different from the required homeworks

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Schedule of readings Before our next exam (October 20th) OpenStax Chapters 1 – 11 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

Homework On class website: Please print and complete homework worksheet #10 Due Thursday October 13th Dan Gilbert Reading and Law of Large Numbers

By the end of lecture today 10/8/15 Use this as your study guide By the end of lecture today 10/8/15 Law of Large Numbers Central Limit Theorem

Review of Homework Worksheet just in case of questions

Homework review 2 5 = .40 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 Based on frequency data (Percent of rockets that successfully launched)

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

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

. .8276 .1056 .2029 .1915 .3944 .4332 .3944 .3944 44 50 55 50 55 52 55 44 - 50 4 = -1.5 52 - 50 4 +.5 55 - 50 4 = = +1.25 z of 1.5 = area of .4332 z of .5 = area of .1915 1.25 = area of .3944 55 - 50 4 55 - 50 4 = +1.25 = +1.25 .5000 - .3944 = .1056 z of 1.25 = area of .3944 z of 1.25 = area of .3944 .4332 +.3944 = .8276 .3944 -.1915 = .2029

Homework review .3264 .2152 .5143 .1255 .3888 .1736 .1736 .3888 3,000 3,000 3,500 2,500 3,500 3000 - 2708 650 2500 - 2708 650 = 0.45 3000 - 2708 650 = 0.45 = -.32 z of 0.45 = area of .1736 z of -0.32 = area of .1255 z of 0.45 = area of .1736 3500 - 2708 650 3500 - 2708 650 = 1.22 = 1.22 .5000 - .1736 = .3264 z of 1.22 = area of .3888 z of 1.22 = area of .3888 .3888 - .1736 = .2152 .3888 +.1255= .5143

Homework review .0764 .9236 .1185 .4236 .4236 .4236 .3051 10 12 20 20 20 - 15 3.5 10 - 15 3.5 = 1.43 20 - 15 3.5 1.43 = -1.43 = z of 1.43 = area of .4236 z of -1.43 = area of .4236 z of 1.43 = area of .4236 12 - 15 3.5 = -0.86 .5000 + .4236 = .9236 .5000 - .4236 = .0764 z of -.86 = area of .3051 .4236 – .3051 = .1185

Comments on Dan Gilbert Reading

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.

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) 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) http://www.youtube.com/watch?v=ne6tB2KiZuk

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 Frequency distributions of individual scores derived empirically we are plotting raw data this is a single sample X Melvin X X X X X X Take a single score Repeat over and over x x x Population x x Preston x x x

important note: “fixed n” 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 Mean for 1st sample

important note: “fixed n” 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

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 Melvin 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

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 X Eugene 23rd sample X 2nd sample

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 Eugene X X X X X X X Melvin X X µ= 100 X X Mean = 100 X X σ = 3 100 Standard Deviation = 3 23rd sample An example of a sampling distribution of sample means 2nd sample µ = 100 Mean = 100 Standard Error of the Mean = 1 = 1 100

Central Limit Theorem x will approach µ 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

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.

population population population n = 2 n = 5 n = 4 n = 30 n = 5 n = 25 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

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

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

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

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

Central Limit Theorem x will approach µ 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

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

Central Limit Theorem x will approach µ 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

Animation for creating 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 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

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 X Eugene 23rd sample X 2nd sample

Central Limit Theorem x will approach µ 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

Writing Assignment: Writing a letter to a friend . 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.

distribution of sample means? 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. .

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Thank you! See you next time!!