Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Spring 2016 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays Welcome

By the end of lecture today 2/26/16 Law of Large Numbers Central Limit Theorem

Homework On class website: Please complete homework worksheet #13 & 14 Homework 13: Dan Gilbert Reading and Law of Large Numbers Homework 14: Letter to a Friend – Central Limit Theorem

Before next exam (March 4th) Schedule of readings Before next exam (March 4th) Please read chapters 1 - 8 in OpenStax textbook 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

Lab sessions Everyone will want to be enrolled in one of the lab sessions Labs continue With Project 2

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

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.

Distribution of raw scores: is an empirical probability distribution Central Limit Theorem 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 Eugene X X X X X X X Melvin X X X X X X Take a single score Repeat over and over x x x Population x x x x

same-sized sample = “fixed n” Central Limit Theorem 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: same-sized sample = “fixed n” Sampling distributions of sample means theoretical distribution we are plotting means of samples Take sample – get mean Repeat over and over Population

Sampling distribution: is a theoretical probability distribution of Central Limit Theorem 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: same-sized sample = “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 23rd sample Sampling distributions sample means theoretical distribution we are plotting means of samples 2nd sample

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 µ = 100 Population Take sample – get mean Repeat Question: What if we made our sample absurdly large, as large as possible?? What if our sample size was HUGE - as large as the whole population? Take sample – get mean Repeat over and over Distribution of means of samples 100 Population µ = 100

Central Limit Theorem µ = 100 Population Take sample – get mean Repeat Question: What if we made our sample absurdly large, as large as possible?? What if our sample size was HUGE - as large as the whole population? What is the variability?? It is zero! Take sample – get mean Repeat over and over Distribution of means of samples 100 Population µ = 100

x will approach µ 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

Homework Assignment 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 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.

Thank you! See you next time!!