1
STAT 2000
Chapter 7
Confidence Intervals
AgrestiFranklin
Identify
Objective
Question(s)
about a
population
Design Study
and
Collect Data
Select a sample
from a population
Make Inferences
The Process of a Statistical Study
Describe
Data
Organize and
present
sample data
about a
population
Make predictions
and draw
conclusions
Principles of
Probability
As we begin to talk about Inference, let’s look back
at what we have done before, and why.
•
Gathering Data – Statistical inference methods
assume that data was collected with some type of
randomization.
•
Sampling Distributions – Probability calculations
Sampling Distributions
Probability calculations
used in inference refer to sampling distributions.
Two types of Inference
•
Estimation – Chapter 7
•
Hypothesis Testing – Chapters 89
A
point estimate
of a population parameter is an
estimate given by a single value.
Sample Statistics are point estimates of
Population Parameters.
Sample
Population
Statistic
Parameter
Mean
μ
Standard Deviation
s
σ
Proportion
p
x
ˆ
p
An
interval estimate
of a population parameter is
given by two values between which we expect to
have the population parameter.
(Why would we want an interval estimate instead of
a point estimate?)
Our
interval estimate
is given by
point estimate
±
margin of error
The
margin of error
is how much we expect to be
*off*.
SAMPLE: Interviews with 706
likely voters, conducted by
telephone on July 911, 2004.
MARGIN
OF
ERROR: ± 4%
The point estimate of the proportion that would vote
for Bush was 0.46.
The interval estimate would be 0.46 ± 0.04, and we
would write this as (0.42, 0.50).
The lower limit is 0.42, and the upper limit is 0.50.
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The
interval estimate
that we will construct is
called a
confidence interval
.
Our confidence is in the method
that we use to
construct this interval.
The
level of confidence
most often used is .95.
We say we have 95% confidence that the interval
we construct contains the parameter.
What we mean is that our method will give correct
results (interval will contain the true population
value) 95% of the time.
In reality, we only take
one sample and
calculate just one
confidence interval.
The diagram to the
right shows what
would happen if we
took many samples.
For each sample, a sample proportion is calculated
and is represented by the center dot.
An interval is
constructed.
The line down the center is the true
population proportion.
The interval constructed
does not always contain the true population value.
When we construct a 95%
confidence interval, we do not
know whether or not the interval
we construct contains the true
population value.
We know that ~ 95%
of the
intervals we could construct
using this same method will
contain
the population value,
and ~ 5% will not.
We say we are 95% confident that our interval
contains the true population proportion.
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 Spring '08
 smith
 Normal Distribution, Standard Deviation, Bush Kerry

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