Transcript Lecture 16

Chapter 7
Statistical Inference:
Confidence Intervals

Learn ….
How to Estimate a Population
Parameter Using Sample Data
Agresti/Franklin Statistics, 1 of 87
 Section 7.1
What Are Point and Interval
Estimates of Population
Parameters?
Agresti/Franklin Statistics, 2 of 87
Point Estimate

A point estimate is a single
number that is our “best guess” for
the parameter
Agresti/Franklin Statistics, 3 of 87
Interval Estimate

An interval estimate is an interval
of numbers within which the
parameter value is believed to fall.
Agresti/Franklin Statistics, 4 of 87
Point Estimate vs Interval
Estimate
Agresti/Franklin Statistics, 5 of 87
Point Estimate vs Interval
Estimate


A point estimate doesn’t tell us how
close the estimate is likely to be to
the parameter
An interval estimate is more useful
• It incorporates a margin of error which
helps us to gauge the accuracy of the
point estimate
Agresti/Franklin Statistics, 6 of 87
Point Estimation: How Do We Make
a Best Guess for a Population
Parameter?

Use an appropriate sample statistic:
• For the population mean, use the sample
•
mean
For the population proportion, use the
sample proportion
Agresti/Franklin Statistics, 7 of 87
Point Estimation: How Do We Make
a Best Guess for a Population
Parameter?

Point estimates are the most common
form of inference reported by the
mass media
Agresti/Franklin Statistics, 8 of 87
Properties of Point Estimators

Property 1: A good estimator has a
sampling distribution that is centered at
the parameter
• An estimator with this property is
unbiased
• The sample mean is an unbiased estimator
of the population mean
• The sample proportion is an unbiased
estimator of the population proportion
Agresti/Franklin Statistics, 9 of 87
Properties of Point Estimators

Property 2: A good estimator has a
small standard error compared to
other estimators
• This means it tends to fall closer than
other estimates to the parameter
Agresti/Franklin Statistics, 10 of 87
Interval Estimation: Constructing an
Interval that Contains the Parameter
(We Hope!)

Inference about a parameter should
provide not only a point estimate but
should also indicate its likely
precision
Agresti/Franklin Statistics, 11 of 87
Confidence Interval


A confidence interval is an interval
containing the most believable values
for a parameter
The probability that this method
produces an interval that contains the
parameter is called the confidence
level
•
This is a number chosen to be close to 1,
most commonly 0.95
Agresti/Franklin Statistics, 12 of 87
What is the Logic Behind
Constructing a Confidence Interval?

To construct a confidence interval for
a population proportion, start with the
sampling distribution of a sample
proportion
Agresti/Franklin Statistics, 13 of 87
The Sampling Distribution of the
Sample Proportion




Gives the possible values for the sample
proportion and their probabilities
Is approximately a normal distribution for
large random samples
Has a mean equal to the population
proportion
Has a standard deviation called the
standard error
Agresti/Franklin Statistics, 14 of 87
A 95% Confidence Interval for a
Population Proportion

Fact: Approximately 95% of a normal
distribution falls within 1.96 standard
deviations of the mean
• That means:
With probability 0.95, the
sample proportion falls within about 1.96
standard errors of the population
proportion
Agresti/Franklin Statistics, 15 of 87
Margin of Error


The margin of error measures how
accurate the point estimate is likely to
be in estimating a parameter
The distance of 1.96 standard errors
in the margin of error for a 95%
confidence interval
Agresti/Franklin Statistics, 16 of 87
Confidence Interval


A confidence interval is constructed
by adding and subtracting a margin of
error from a given point estimate
When the sampling distribution is
approximately normal, a 95%
confidence interval has margin of
error equal to 1.96 standard errors
Agresti/Franklin Statistics, 17 of 87
 Section 7.2
How Can We Construct a
Confidence Interval to Estimate a
Population Proportion?
Agresti/Franklin Statistics, 18 of 87
Finding the 95% Confidence Interval
for a Population Proportion



We symbolize a population proportion by p
The point estimate of the population
proportion is the sample proportion
We symbolize the sample proportion by pˆ
Agresti/Franklin Statistics, 19 of 87
Finding the 95% Confidence Interval
for a Population Proportion


A 95% confidence interval uses a margin of
error = 1.96(standard errors)
[point estimate ± margin of error] =
pˆ  1.96(standard errors)
Agresti/Franklin Statistics, 20 of 87
Finding the 95% Confidence Interval
for a Population Proportion

The exact standard error of a sample proportion
equals:
p(1  p)
n


This formula depends on the unknown population
proportion, p
In practice, we don’t know p, and we need to
estimate the standard error
Agresti/Franklin Statistics, 21 of 87
Finding the 95% Confidence Interval
for a Population Proportion

In practice, we use an estimated standard
error:
se 
p
ˆ (1  p
ˆ)
n
Agresti/Franklin Statistics, 22 of 87
Finding the 95% Confidence Interval
for a Population Proportion

A 95% confidence interval for a population
proportion p is:
ˆ  1.96(se), with se 
p
ˆ (1 - p
ˆ)
p
n
Agresti/Franklin Statistics, 23 of 87
Example: Would You Pay Higher
Prices to Protect the Environment?

In 2000, the GSS asked: “Are you
willing to pay much higher prices in
order to protect the environment?”
• Of n = 1154 respondents, 518 were
willing to do so
Agresti/Franklin Statistics, 24 of 87
Example: Would You Pay Higher
Prices to Protect the Environment?

Find and interpret a 95% confidence
interval for the population proportion
of adult Americans willing to do so at
the time of the survey
Agresti/Franklin Statistics, 25 of 87
Example: Would You Pay Higher
Prices to Protect the Environment?
518
pˆ 
 0.45
1154
(0.45)(0.55)
se 
 0.015
1154
pˆ  1.96(se) 1.96(0.015)
 0.45 0.03 (0.42,0.48)
Agresti/Franklin Statistics, 26 of 87
Sample Size Needed for Large-Sample
Confidence Interval for a Proportion

For the 95% confidence interval for a
proportion p to be valid, you should have at
least 15 successes and 15 failures:
ˆ )  15
np
ˆ  15 and n(1 - p
Agresti/Franklin Statistics, 27 of 87
“95% Confidence”


With probability 0.95, a sample
proportion value occurs such that the
confidence interval contains the
population proportion, p
With probability 0.05, the method
produces a confidence interval that
misses p
Agresti/Franklin Statistics, 28 of 87
How Can We Use Confidence
Levels Other than 95%?



In practice, the confidence level 0.95
is the most common choice
But, some applications require
greater confidence
To increase the chance of a correct
inference, we use a larger confidence
level, such as 0.99
Agresti/Franklin Statistics, 29 of 87
A 99% Confidence Interval for p
pˆ  2.58(se)
Agresti/Franklin Statistics, 30 of 87
Different Confidence Levels
Agresti/Franklin Statistics, 31 of 87
Different Confidence Levels

In using confidence intervals, we
must compromise between the
desired margin of error and the
desired confidence of a correct
inference
• As the desired confidence level
increases, the margin of error gets
larger
Agresti/Franklin Statistics, 32 of 87
What is the Error Probability for
the Confidence Interval Method?

The general formula for the confidence
interval for a population proportion is:
Sample proportion ± (z-score)(std. error)
which in symbols is
pˆ  z(se)
Agresti/Franklin Statistics, 33 of 87
What is the Error Probability for
the Confidence Interval Method?
Agresti/Franklin Statistics, 34 of 87
Summary: Confidence Interval
for a Population Proportion, p

A confidence interval for a population
proportion p is:
ˆ z
p
ˆ (1 - p
ˆ)
p
n
Agresti/Franklin Statistics, 35 of 87
Summary: Effects of Confidence
Level and Sample Size on Margin of
Error

The margin of error for a confidence
interval:
• Increases as the confidence level
increases
• Decreases as the sample size
increases
Agresti/Franklin Statistics, 36 of 87
What Does It Mean to Say that
We Have “95% Confidence”?

If we used the 95% confidence
interval method to estimate many
population proportions, then in the
long run about 95% of those intervals
would give correct results, containing
the population proportion
Agresti/Franklin Statistics, 37 of 87
A recent survey asked: “During the
last year, did anyone take something
from you by force?”


a.
b.
c.
Of 987 subjects, 17 answered “yes”
Find the point estimate of the proportion
of the population who were victims
.17
.017
.0017
Agresti/Franklin Statistics, 38 of 87