Statistics for Social
and Behavioral Sciences
Session #16:
Confidence Interval and Hypothesis Testing
(Agresti and Finlay, from Chapter 5 to Chapter 6)
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’s ratings are within a MoE of each other!
PART IV. : CORRELATION AND CAUSATION:
REGRESSION ANALYSIS
This is where we talk
about Zmapp and Ebola!
Weeks 10-14
Last Session
• Interval estimates for a population mean
(a parameter) when N is large, for any distribution of X.
• Build a confidence interval for a parameter:
the interval [Sample Mean – MoE ; Sample + MoE]
includes the parameter with probability:
99% if MoE = 2.58 * Standard Error
95% if MoE = 1.96 * Standard Error
90% if MoE = 1.65 * Standard Error
Today
• What happens if N is small?
– The Central Limit Theorem does not apply.
– If X is normally distributed, then we have a way of
getting the sampling distribution of the sample
mean…. And thus build confidence intervals,
standard errors.
– The sample mean follows a t-distribution.
• Hypothesis testing in Statistics:
the Foundation of (Social) Sciences.
– The Confidence Interval method of testing m=v.
Outline
1. Small samples, normal distribution:
t-distribution
2. Testing hypothesis:
The Foundation of (Social) Sciences
Next time:
t test of mean and proportion
Chapter 6 of A&F
Central Limit Theorem
• Requires a large sample size N.
• This is because it applies to any distribution of X.
• Example:
– We had a sample of N songs, and the number of times
Xi that song had been played.
– The number of times Xi a song is played on Spotify does
not have a normal distribution.
– But we can build a confidence interval for the average
number of times a song is played (m), provided we have
a large enough number N of songs.
– MoE = 1.96 * sX/√N for a 95% confidence
interval.
N is large !! The distribution of X may be
different from a normal distribution.
We can use our formulas to find
a 95% confidence interval for
m=360.63 as:
• N is large.
Even though X does not have a
normal distribution.
What if N is small?
• If N is “small”, the Central Limit Theorem does
not apply….
– We cannot use our formulas.
• “Small” ? Less than a few hundred (from
experience).
• If N is very small:
N=2
These sampling
distributions are not
normal.
N=5
If N is small
• sX is potentially very far from sx.
• But… we can still find confidence intervals if X is normal.
• The sampling distribution of the sample mean is Student’s
t distribution, with degrees of freedom (df) equal to N-1,
and with standard deviation sx/√N.
If N is small
A 95% confidence interval for the sample mean is:
[Sample Mean – MoE , Sample Mean + MoE]
With MoE = z * Standard Error.
• z= 1.96
when the df = ∞
• z> 1.96
when the df are small.
• See next table for the exact value of z.
t Table
Why is it called Student’s t
distribution?
• The t distribution was allegedly invented by a person
called Student.
• That “Student” was an engineer at factories in
Ireland: William Sealy Gossett.
• He was producing small samples of a p, seeking
guidance for industrial quality control:
– He was trying a small number of samples
(N=2,4, perhaps 7).
– And from these samples was trying to infer the quality of
all containers of the product (the population).
W.S. Gosset and Some Neglected Concepts in Experimental Statistics, Stephen T. Ziliak,
2011.
Outline
1. Small samples, normal distribution:
t-distribution
2. Testing hypothesis:
The Foundation of (Social) Sciences
Next time:
t test of mean and proportion
Chapter 6 of A&F
Thinking like a statistician: Step 1
• Empirical question, type #1:
“ What is an estimate of the population proportion
of voters for Republicans in Colorado?”
Or…
• Empirical question, type #2:
In Plain English
“Is Cory Gardner likely to win the election?”
In more formal terms:
“Can we reject the hypothesis that Cory Gardner
will lose the election with some confidence?”
Thinking like a statistician: Step 1
• Empirical question, type #1:
“ What is an estimate of the population impact of
Zmapp on Ebola ?”
Or…
• Empirical question, type #2:
In Plain English
“Is Zmapp effective at treating Ebola?”
In more formal terms:
“Can we reject the hypothesis that Zmapp does
not treat Ebola patients with some confidence?”
Hypothesis testing
• Hypothesis: an empirical statement about a population
parameter. Usually of the shape:
– “The parameter is equal to a given value”
– “The parameter is greater than a given value”
– “The parameter is lower than a given value”
• Almost all scientific/sociological/economic statements
can be reduced to one of these three types.
– “The population proportion of voters for Cory Gardner is
greater than 50%.” (second type of hypothesis)
– “The impact of ZMapp on Ebola patients’ condition is zero.”
(first type of hypothesis)
Hypothesis
• Fundamental principle of scientific analysis: we can
only provide evidence to reject a hypothesis. We
never actually accept a hypothesis…
• “Science must begin with myths, and with the
criticism of myths.” Karl Popper.
• Scientific hypothesis are falsifiable, i.e.
it is possible to bring data to test such hypothesis.
• This applies to social science as well: impact of taxes
on individuals’ mobility, impact of abortion on crime
(Steve Levitt, Freakonomics).
• Logik der Forschung, Vienna, 1935.
• Translated into The Logic of Scientific Discovery,
1959.
Formulating Hypothesis
• H0 Null hypothesis (to be rejected by evidence): “Zmapp
does not improve Ebola patients’ condition.” or “The
population impact of Zmapp on Ebola is zero.”
• Ha Alternative hypothesis: “Zmapp has a positive impact
on the population of Ebola patients’ condition.”
See that the formulation of the
alternative matters. Beware of
your priors.
Formulating Hypothesis (simpler)
• H0 Null hypothesis (to be rejected by evidence):
“The fraction of men in Abu Dhabi is equal to
50%.”
• Ha Alternative hypothesis: “The fraction of men
in Abu Dhabi is different from 50%.”
A tourist’s pic. What’s going on?
See that the formulation of the
alternative matters. Beware of
your priors.
Testing H0: m=v using confidence intervals
• H0: “The fraction of men in Abu Dhabi is 50%.” equivalently “m = 0.5”.
• By simple random sampling, gather N observations Xi=0,1.
• Build a confidence interval for the sample mean m of Xi.
– Same methods as seen in previous sessions.
• If the null hypothesis is true, only 5 of the 95% confidence intervals will
not include 0.5.
• Thus if the null hypothesis is true, there is only a 5% probability that my
confidence interval will not include 0.5.
☞ Reject the null hypothesis if the confidence interval for m does not
include v.
Statistical Errors:
“The Truth (m) is out there”
Null hypothesis is true
Alternative hypothesis
is true
Do not reject the null
hypothesis
Correct decision
Type II error
We reject the null
hypothesis
Type I error
Correct decision
• By selecting 95% confidence intervals, what is the probability of a type I error?
• This is called the significance level (a level) of the test.
• It is not possible to make no type I and no type II error.
Wrap up
• When the sample size is small: The t-distribution gives confidence
intervals … and when the distribution of X is normal.
• Use z given by Table 5.1 of Agresti and Finlay for degrees of
freedom N-1.
Then MoE = z * Standard Error
• Hypothesis testing is the foundation of (social) sciences.
• Hypothesis: A parameter is equal to …. , A parameter is greater
than …., A parameter is lower than …. .
• We can only provide evidence to reject a null hypothesis.
• Confidence interval method for the test of H0 : m = v. Ha: m ≠ v.
– Reject the H0 with significance level 5% if the 95% confidence interval for the
sample mean m does not include v.
– Reject the H0 with significance level 10% if the 90% confidence interval for
the sample mean m does not include v.
Coming up:
Readings:
• Mid term on Tuesday, November 25.
– Coverage: up to Chapter 6.
•
Deadlines are sharp and attendance is followed.
For help:
• Amine Ouazad
Office 1135, Social Science building
amine.ouazad@nyu.edu
Office hour: Tuesday from 5 to 6.30pm.
• GAF: Irene Paneda
Irene.paneda@nyu.edu
Sunday recitations.
At the Academic Resource Center, Monday from 2 to 4pm.