Stage Screen Row B Gallagher Theater Row R Lecturer’s desk Row A Row B Row C Row A Row A Row C Row D Row D Row E Row E Row F Row F Row G Row G Row H Row H Row I Row I Row J Row J Row K Row K Row L Row L Row M Row M Row N Row N Row O Row O Row P Row P Row Q Row Q 4 4 Row R 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

Stage Screen 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 R/L handed broken desk Social Sciences 100 Row N Row O Row P Row Q Row R 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 Row I Lecturer’s desk 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 Row B

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

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

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

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

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

Review of Homework Worksheet just in case of questions

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)

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 Based on frequency data (Percent of times at bat that successfully resulted in hits)

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

= = z of 1.5 = area of =.8276 z of 1.25 = area of = = = area of = = z of.5 = area of =.2029 z of 1.25 = area of

Homework review = 0.45 z of 0.45 = area of = = 0.45 z of 0.45 = area of = = 1.22 z of 1.22 = area of = -.32 z of = area of = = 1.22 z of 1.22 = area of

Homework review = 1.43 z of 1.43 = area of = = 1.43 z of 1.43 = area of.4236 z of = area of –.3051 =.1185 z of -.86 = area of = = =

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)

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

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

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”

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

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

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 = 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

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

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.

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

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 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.

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

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.

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

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 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

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

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

. 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.

. 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.