Download Slides for Session #12

Statistics for Social
and Behavioral Sciences
Session #12:
Sampling Distribution
Central Limit Theorem
(Agresti and Finlay, Chapter 9)
Prof. Amine Ouazad
Statistics Course Outline
PART I. INTRODUCTION AND RESEARCH DESIGN
Week 1
PART II. DESCRIBING DATA
Weeks 2-4
PART III. DRAWING CONCLUSIONS FROM DATA:
INFERENTIAL STATISTICS
Weeks 5-9
Firenze or Lebanese Express is coming up next session
PART IV. : CORRELATION AND CAUSATION:
REGRESSION ANALYSIS
This is where we talk
about Zmapp and Ebola!
Weeks 10-14
Last Session
• A random variable is a variable whose value has not been
realized.
• The expectation of a random variable Y is:
E(Y) = S yk P(Y=yk)
Also, E(X+Y) = E(X) + E(Y),
and E(c X)=c E(X), and E(E(X|Z))=E(X)
• Typically the probability distribution P is not known, but we
approximate it….
– Using the distribution for past values of Y (example: earnings of
previous graduates)
– Using polls, to ask individuals for example how they will vote.
• The normal distribution is an ubiquitous distribution, that is
symmetric, bell shaped. It is characterized by its mean m and its
standard deviation s.
• The standard normal distribution has mean 0 and standard
deviation 1.
Outline
1. The standard normal distribution
Z-Score
2. Polls and normal distributions
Sampling distribution of a statistic
A simulation
Central Limit Theorem
Variance of a yes/no (dummy) variable
Next time:
Probability Distributions (continued)
Chapter 4 of A&F
Comparing test scores across colleges
“Early paleontology in Indianapolis”
“Hip hop in the Middle East”
Test scores have a normal
distribution with mean 3
and standard deviation 4.
Test scores have a normal
distribution with mean 4
and standard deviation 1.
• Problem: how do I compare Marina’s test score of 3.6 at the paleontology course
with a test score of 4.1 at the Hip Hop in the Middle East?
Z-score !
• Take a student’s paleontology test score at the
end of the semester. This is a random variable.
– Its probability distribution has a mean of m=3 with
a standard deviation of s=4.
– Now consider the “z-scored” paleontology test
score:
z - scored paleontology =
paleontology test score - m
– The z-scored paleontology test scoreshas a mean
of 0, and a standard deviation of 1.
Standard Normal Distribution
• Is simply the normal distribution with mean 0
and standard deviation 1.
• A z-score of 3 means that the student is three
times the standard deviation (of original test
scores) above the mean.
So who has a better grade, Marina
or Slavoj?
Outline
1. The standard normal distribution
Z-Score
2. Polls and normal distributions
Sampling distribution of a statistic
A simulation
Central Limit Theorem
Variance of a yes/no (dummy) variable
Next time:
Probability Distributions (continued)
Chapter 4 of A&F
Who will win the mid term
elections in the US?
• Mid term elections are held two years after
the presidential elections in the United States.
• They take place early november 2014.
• A question: what fraction of the voters will
vote for Cory Gardner in Colorado?
Mario Zapata Encinas
Real Clear Politics: Gardner vs. Udall
- Who will win?
It would be logical to think that Gardner will win, because
from the statistics, he has a higher percentage of votes
(without taking into consideration the margin of error of
these statistics)
- What is MoE?
MoE stands for Margin of Error, which is a statistic
expressing the random sampling error in survey results
-
What is the likely distribution of the fraction of
voters who will vote for Gardner?
According to RCP, 46.6/100 of voters will choose Gardner
over Udall.
Colorado Senate: Gardner (R) vs Udall (D)
The average MoE is 3.68%
Thus, the likely distribution of Gardner voters is
between 43.42% to 50.78%
LIKELY
I think Gardner will win the election.
WINNER
Nick Chaubey
10-28
Conducting a poll
• Goal: estimating the fraction of individuals who
intend to vote for a candidate.
• Thinking like a statistician:
1. Ask an empirical question.
2. Design the study: population? Sample? Sampling
method? Response, nonresponse bias?
3. Describe the data: what is the mean of the sample?
4. Make inferences: in plain English, can we predict what
candidate will win?
Polling company methodology
• Select a sample of individuals by either simple
random sampling, cluster sampling, or
stratified random sampling.
• Sample size N.
• Ask each individual i=1,2,…,N:
– « Which candidate do you intend to vote for? »
– Note VoteGardneri=1 if individual i intends to vote
for Gardner, and 0 otherwise.
• Report the mean of the sample:
1 N
Mean(VoteGardner) = åVoteGardneri
N i=1
But there is sampling error!
Groundhog Day: In expectation
• Take the i-th individual that will be contacted by
Rasmussen.
• The probability that « individual i-th declares
voting for Gardner » (an event) is
– The true fraction of individuals who intend to vote for
Gardner in the US population of eligible voters.
• Write VoteGardneri=Xi the random variable:
• 1 if the i-th individual declares intending to vote for Gardner,
• and 0 otherwise.
• Then:
E(Xi) = true fraction of individuals who will vote for Gardner.
We write it E(Xi) = p
Groundhog Day: In Expectation
• Now what is the expected value of the fraction
declaring they will vote for Gardner?
1 N
Mean(VoteGardner) = åVoteGardneri
N i=1
• It is:
1 N
E ( Mean(VoteGardner)) = E( åVoteGardneri )
N i=1
• Now remember that E(X+Y)=E(X)+E(Y) so…
E(Mean(VoteGardner)) = p
• The polling company will get the true fraction of
voters for Gardner… in expectation!
Sampling distribution of a statistic
• But there is some chance that the mean will be
far off the true fraction… what probability?
• A statistic is a random variable.
• Indeed the % of respondents who say they intend
to vote for Gardner depends on the sample that
was drawn.
• This is random as the sample was collected by
simple random sampling.
• The mean is a random variable:
1 N
E ( Mean(VoteGardner)) = E( åVoteGardneri )
N i=1
Central Limit Theorem
That is the probability
that the reported
fraction is equal to 30%
Probability(Mean(VoteGardner)=m)
With some (low)
probability the polling
company will give a
number ‘far’ over the
true fraction of voters
for Gardner
The reported fraction
could be here, e.g. 30%
With
probability
95%,
the
estimated fraction of voters for
Gardner will be between the true
fraction + - 2 standard deviations of
the distribution.
True fraction of voters for Gardner
With some (low)
probability the polling
company will give a
number ‘far’ over the
true fraction of voters
for Gardner
m
• Central Limit Theorem: With a large sample size, the sampling
distribution of the mean(VoteGardner) is normal, and the empirical
rule applies.
Central Limit Theorem
• The last remaining element is the standard deviation of the
sampling distribution.
• Noting sX the standard deviation of X, the sampling
distribution of the mean of X has standard deviation:
sX
N
• The standard deviation of the sampling distribution is called
the standard error. It is a measure of sampling error.
• Finally what is sX?
• For a proportion, sX = √( p (1-p) ) , where p is the true
value.
Good news
• There is some probability that the reported mean will be far above
or far below the true mean. But:
• With a large sample size, the probability that the reported mean
is further than 2 standard errors from the true mean is 5%.
• The most likely outcome is the true mean.
– The mode of the sampling distribution is the true mean.
• The expected value of the reported mean is the true
mean.
• The larger the sample size, the smaller the standard
error.
Bad news
• We measure the reported mean, we know the
sample size….
• But we don’t know the true mean.
• Without the true mean we cannot know what
the sampling distribution is…
– we miss both the mean (p) and the standard
deviation ( sX / √N ) (aka standard error) of that
statistic.
• If we knew p, the true mean, there would be
no need for a poll.
Next session: the solution to this conundrum
Exercise: Compute the
Margin of Error
• The Rasmussen Poll interviewed 966
individuals.
• Assuming that the true fraction of individuals
who will vote for Gardner is 50%, what is the
Margin of Error?
• The Margin of Error is here two standard
errors of the distribution.
• Is it close to the result reported by the
website?
Wrap up
• A statistic is a random variable.
• The distribution of a statistic is called its sampling
distribution.
• In particular the mean of a variable in a sample is a
statistic.
• The expected value of the sample mean is equal to
the true mean.
• The standard deviation of the sample mean is called
the standard error.
• Central Limit theorem: with a large sample size, the
sampling distribution of the mean of X is normal, and
the empirical rule applies.
The standard error is sX / √N.
Coming up:
Readings:
• Chapter 5 entirely – estimation, confidence intervals.
• Online quiz on Thursday.
For help:
• Amine Ouazad
Office 1135, Social Science building
[email protected]
Office hour: Tuesday from 5 to 6.30pm.
• GAF: Irene Paneda
[email protected]
Sunday recitations.
At the Academic Resource Center, Monday from 2 to 4pm.