1. 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

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

4. 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

5. Before next exam (March 4th)
Schedule of readings
Before next exam (March 4th)
Please read chapters 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

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

10. 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)

11. 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”)

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

13. .
.8276
.1056
.2029
.1915
.3944
.4332
.3944
.3944 44 50 55 50 55 52 55 4 =
-1.5 4
+.5 4 = =
+1.25
z of 1.5 = area of .4332
z of .5 = area of .1915
1.25 = area of .3944 4 4 =
+1.25 =
+1.25
= .1056
z of 1.25 = area of .3944
z of 1.25 = area of .3944
= .8276
= .2029

14. Homework review
.3264
.2152
.5143
.1255
.3888
.1736
.1736
.3888
3,000
3,000 3,500
2, ,500
650
650 =
0.45
650 =
0.45 =
-.32
z of 0.45 = area of .1736
z of = area of .1255
z of 0.45 = area of .1736
650
650 =
1.22 =
1.22
= .3264
z of 1.22 = area of .3888
z of 1.22 = area of .3888
= .2152
= .5143

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

16. 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.

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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.

26. Thank you!
See you next time!!