Download Slides for Session #15

Statistics for Social
and Behavioral Sciences
Session #15:
Interval Estimation, Confidence Interval
(Agresti and Finlay, Chapter 5)
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: Inference
Central Limit Theorem:
• with a large sample size N, the sampling distribution of the sample mean is
approximately normal.
• The mean of the sampling distribution is the population mean.
• The standard deviation of the sampling distribution is sX/√N, where sX is the
standard deviation of X.
• A conservative Margin of Error (= 2 standard errors)
for Cafe Firenze’s restaurant rating is 1.1 with 14
votes.
• For any rating from 1 to 5, the largest possible Margin
of Error is 4/√N, where N is the number of ratings.
• With TripAdvisor, we see the rating of each individual
customer, and so we can calculate sX!
Today
• Use this margin of error to provide interval
estimates:
– A 95% confidence interval for Café Firenze is [2.3,4.5].
– “The true rating of Café Firenze is between 2.3 and 4.5
with probability 95%”.
– Note: average was 3.4 and MoE was 1.1.
– A 95% confidence interval for Cory Gardner’s vote share
in Colorado is [48-3.6,48+3.6]=[44.4,51.6].
– “The true vote share for Cory Gardner is between
42.9% of the vote and 50.1% of the vote with 95%
probability”.
– Note: MoE was 3.6.
News: Last Tuesday
• We learnt the population proportion p !!!
– Proportion of voters for Cory Gardner.
• The latest poll was giving us a
sample proportion of the vote p (N around 1000).
Outline
1. Interval Estimation
Confidence Interval
2. Choosing between 90, 95, 99% confidence
3. When distributions are normal: t-distribution
Next time:
Estimation, Confidence Intervals (continued)
Chapter 5 of A&F
Parameters and Interval Estimate
• An interval estimate is an interval of numbers
around the point estimate, which includes the
parameter with probability either 90%, 95%,
or 99%.
• Example:
“the interval estimate
[156.2 cm – 0.49cm ; 156.2 cm + 0.49cm]
includes the population average height with probability 95%.”
• Sample mean: 156.2cm, MoE = 0.49 cm.
Parameters and Interval Estimate
• An interval estimate that includes the parameter with
probability 95% is called a 95% confidence interval.
• The expression “95% confidence interval” is widely used.
• Example:
“[156.2 cm – 0.49cm ; 156.2 cm + 0.49cm]
is a 95% confidence interval for the population average height.”
• Sample mean: 156.2cm, MoE = 0.49 cm.
We use 1.96 instead
of 2 from now on.
How do we build a
95% confidence interval?
Goal: estimate the population average m.
From previous sessions:
[m – MoE ; m + MoE] includes the sample mean with
probability 95%.
We conclude: the interval [m – MoE; m+MoE] includes the
population mean with probability 95%.
[m – MoE; m+MoE] is a 95% confidence interval for m.
MoE = 1.96 x Standard Error
Standard Error = sX/√N
Outline
1. Interval Estimation
Confidence Interval
2. Choosing between 90, 95, 99% confidence
3. When distributions are normal: t-distribution
Next time:
Estimation, Confidence Intervals (continued)
Chapter 5 of A&F
Choosing between 90%, 95%, 99%
• The interval estimate
[Sample Mean – MoE, Sample Mean + MoE]
includes the population mean (the parameter)
with probability:
• 99% if MoE = 2.58
• 95% if MoE = 1.96
• 90% if MoE = 1.65
* Standard Error
* Standard Error
* Standard Error
• The width of a confidence interval:
1. Increases as the confidence level increases.
2. Decreases as the sample size increases.
Building 90%, 95%, 99%
confidence intervals
Exercise:
• The sample mean weight (a sample of
individuals in the US) is 60.0 kg, and the
sample standard deviation is 29.9 kg.
• Find a 90% (resp., 95%, 99%) confidence
interval for the population mean weight.
Why 90%, 95%, 99%?
• Invented by Jerzy Newman in the 1930s.
• R.A. Fisher developed the theory of
statistical testing.
• Sample sizes were small at the time
(a few hundred), and 95% seemed
a reasonable confidence level.
• Medical sciences introduced
confidence intervals in medicine
soon after their discoveries.
• 95% became the standard.
R.A. Fisher
Outline
1. Interval Estimation
Confidence Interval
2. Choosing between 90, 95, 99% confidence
3. When distributions are normal: t-distribution
Next time:
Estimation, Confidence Intervals (continued)
Chapter 5 of A&F
Central Limit Theorem
• Requires a large sample size N.
• This is because it applies to any distribution of X.
• Example #1:
– 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.
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 Guinness’s
Factories in Ireland: William Sealy Gossett.
• He was producing small samples of a drink, 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: Guinnessometrics II,
Stephen T. Ziliak, 2011.
Wrap up
• 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
• The t-distribution gives confidence intervals when the sample size
N is small… 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.
Coming up:
Readings:
• This week and next week:
– Chapter 5 entirely – estimation, confidence intervals.
• Online quiz deadline Tuesday 9am.
•
Deadlines are sharp and attendance is followed.
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.