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Introduction to Statistics for the Social Sciences
SBS200, COMM200, GEOG200, PA200, POL200, or SOC200Lecture Section 001, Fall 2015
Room 150 Harvill Building10:00 - 10:50 Mondays, Wednesdays & Fridays.
http://courses.eller.arizona.edu/mgmt/delaney/d15s_database_weekone_screenshot.xlsx
Everyone will want to be enrolled in one of the lab sessions
Labs continue next week
Please re-register your clickerhttp://
student.turningtechnologies.com/
By the end of lecture today10/9/15
Law of Large Numbers
Central Limit Theorem
Before next exam (October 16th)
Please read chapters 1 - 8 in OpenStax textbook
Please read Chapters 10, 11, 12 and 14 in PlousChapter 10: The Representativeness Heuristic
Chapter 11: The Availability HeuristicChapter 12: Probability and Risk
Chapter 14: The Perception of Randomness
Schedule of readings
On class website: Please print and complete homework worksheet #11 Due Monday October 12th
Dan Gilbert Reading and Law of Large Numbers
Homework
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
25
= .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”)
30100
= .30
Based on frequency data (Percent of times at bat that successfully resulted in hits)
Homework review5
50= .10
Based on frequency data (Percent of students who successfully chose to be Economics majors)
.
50 554444 - 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
50 55
55 - 50 4
= +1.25
.5000 - .3944 = .1056
1.25 = area of .3944
.3944
52 5552 - 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
Homework review
3,0003000 - 2708
650 =0.45
z of 0.45 = area of .1736
.5000 - .1736 = .3264
3,000 3,500
.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
2,500 3,500
.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
Homework review
20 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
10 1220
.4236
.5000 + .4236 = .9236
10 - 15 3.5 =-1.43
12 - 15 3.5 =-0.86
.0764
.1185
.9236
.3051.4236
Comments on Dan Gilbert Reading
Law of large numbers: As the number of measurementsincreases 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 measurementsincreases 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)
Sampling distributions of sample means versus frequency distributions of individual scores
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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 xx
xx
xxx
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-sizedsamples 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 1st 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-sizedsamples 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-sizedsamples from a population
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2nd sample
23rd 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 drawnfrom any population (regardless of the shape of thepopulation distribution), as N becomes larger, the distribution of sample means approaches normality, with the overall mean approaching the theoretical populationmean.
Sampling distribution for continuous distributions
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MelvinEugen
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Sampling Distribution of Sample means
Distribution of Raw Scores
2nd sample
23rd sample
An example of asampling distribution of sample means
µ= 100σ = 3
= 1
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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-sizedsamples from a population
Mean = 100
100
100
Standard Deviation = 3
µ = 100Mean = 100
Standard Errorof the Mean = 1
Notice: SEM is smaller than SD – especially as n increases
Melvin
Eugene
2nd sample
23rd 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
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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