Sections 7.1 and 7.2
Point estimate: a single number
Interval estimate: interval of numbers.
This chapter presents the beginning
of inferential statistics.
The two major applications of inferential
statistics
Estimate
Test
a population parameter: proportion, mean
some claim (or hypothesis) about a population.
Point Estimate
Confidence Interval
p=
Why?: point estimate is not reliable under
rere-sampling.
A confidence interval (CI): an interval of
values used to estimate the true population
parameter.
pˆ = nx
(pronounced
‘p-hat’)
population proportion
sample proportion
of x successes in a sample of size n.
Unbiased estimate (best estimate)
qˆ = 1 - pˆ = sample proportion
of failures in a sample size of n
Confidence Level
Example: PhotoPhoto-Cop Survey Responses
829 adult Minnesotans were surveyed, and 51% of them are
opposed to the use of the photophoto-cop for issuing traffic
tickets. Using these survey results, find the best estimate of
the proportion of all adult Minnesotans opposed to photophotocop use.
Best point estimate=sample proportion=51%.
α: between 0 and 1
A confidence level: 1 - α or 100(1100(1- α)%. E.g. 95%.
This is the proportion of times that the confidence
interval actually does contain the population parameter,
assuming that the estimation process is repeated a
large number of times.
Other
names: degree of confidence or the
confidence coefficient.
coefficient.
1
The Critical Value
(z(z-score)
Finding zα/2
100(1- α)% Confidence Level
α/2 for 100(1-
Given α
α =5%
α/2
α/ = 2.5% = .025
Margin of Error of ^
p
Sampling Distribution of ^
p
The sampling distribution of sample proportion can be
approximated by a normal distribution if np≥
np≥15 and
nq ≥15 : phat is approximately N(p, pq/n),
pq/n), q=1q=1-p.
p
z=
the maximum likely (with probability 1 – α)
difference between the observed proportion
^ and the true population proportion p.
p
pˆ − p
pˆ qˆ
n
E = zα / 2
ˆp q̂
n
^
Standard Error of p
=se
^
p
p
Finding the 95% Confidence Interval
for a Population Proportion
A 95% confidence interval for a population
proportion p is:
p̂ ± 1.96(se), with se =
with
p̂(1 - p̂)
n
100(1100(1-α)% confidence interval for p is
pˆ ± zα / 2 ( se)
Example: Would You Pay Higher
Prices to Protect the Environment?
se =
Of
n = 1154 respondents, 518 were willing to
do so
pˆ (1 − pˆ )
n
In 2000, the GSS asked: “Are you willing to
pay much higher prices in order to protect
the environment?”
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
2
Example: Would You Pay Higher
Prices to Protect the Environment?
What is the Error Probability for the
Confidence Interval Method?
518
= 0.45
1154
(0.45)(0.55)
se =
= 0.015
1154
E = 1.96(se) = 1.96(0.015) = 0.03
p̂ ± E = 0.45 ± 0.03 = (0.42, 0.48)
p̂ =
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
Determining Sample Size
Recall :
E=
zα / 2
pˆ qˆ
n
(solve for n by algebra)
2
n = ( zα /E22) pˆ qˆ
Sample Size for Estimating
Proportion p
ˆ
When an estimate p of p is known:
2
n = ( zα / E2 )2 pˆ qˆ
When no estimate of p is known:
Example:
Example: Suppose a sociologist wants to determine
the
current percentage of U.S. households using ee-mail. How many
households must be surveyed in order to be 95% confident that the
sample percentage is in error by no more than four percentage
points?
points?
a) Use this result from an earlier study: In 1997, 16.9% of U.S.
U.S.
households used ee-mail (based on data from The World Almanac
and Book of Facts).
b) Assume that we have no prior information suggesting a possible
possible
value of p.
n = ( zα / E2)2
2 0.25
3
b) Assume that we have no prior information suggesting a possible
possible
value of p.
a) Use this result from an earlier study: In 1997, 16.9% of U.S.
U.S.
households used ee-mail (based on data from The World Almanac
and Book of Facts).
n = [za/2 ]2ˆpˆq
E2
= [1.96]2 (0.169)(0.831)
0.042
= 337.194
= 338 households
To be 95% confident that our
sample percentage is within
four percentage points of the
true percentage for all
households, we should
randomly select and survey
338 households.
n = [za/2 ]2 • 0.25
E2
= (1.96)2 (0.25)
0.042
= 600.25
= 601 households
With no prior information,
we need a larger sample to
achieve the same results
with 95% confidence and an
error of no more than 4%.
Finding the Point Estimate
and E from a
Confidence Interval
Point estimate of p:
ˆ
p = (upper confidence limit) + (lower confidence limit)
2
Margin of Error:
E = (upper confidence limit) — (lower confidence limit)
2
4