Transcript Chapter 19: Confidence Intervals for Proportions “Far better an approximate answer to the right question,…than an exact answer to the wrong question.” -John W.

Chapter 19:
Confidence Intervals for
Proportions
“Far better an approximate answer
to the right question,…than an exact
answer to the wrong question.”
-John W. Tukey
Standard Error
 To find the standard error:
 
SE p 
pq
n
 Because the sampling distribution model is
Normal:
68% of all samples will be within
p  1SE
95% of all samples will be within
p  2SE
99.5% of all samples will be within
p  3SE
Confidence Interval
 “One-proportion
z-interval”
Putting a number
on the probability
that this interval
covers the true
proportion.
Our best guess of
where the
parameter is and
how certain we
are that it’s within
some range.
Margin of Error
 The extent of the interval on either side of
p is called the margin of error (ME).
 In general, confidence intervals are written as:
estimate  ME
 There is a conflict between certainty and
precision
Choose a confidence level – the data does not
determine the confidence level
Assumptions and Conditions
 Independence Assumption:
The data values are assumed to be independent
from each other.
Plausible independence condition:
 Do the data values somehow affect each other?
 Dependent on knowledge of the situation
Randomization condition:
 Where data sampled at random or generated from a properly
randomized experiment?
 Proper randomization helps ensure independence
10% condition:
 Samples are always drawn without replacement
 Samples size should be less than 10% of the population
Assumptions and Conditions
Sample Size Assumption:
-Based upon the Central Limit Theory (CLT)
The sample must be large enough to make the
sampling model for the sampling proportions
approximately Normal.
More data is needed as the proportion gets closer
to either extreme, 0 or 1.
Success/failure condition: expect at least 10
successes and 10 failures.
.
One-proportion z-interval
 When the conditions are met, we are ready to find
the confidence interval for the population
proportion, p. Since the standard error of the
proportion is estimated by
 
SE p 
pq
, the confidence interval is
n
 
p  z  SE p . The critical value, z , depends
*
*
on the particular confidence level, C, that you specify
TI-83+ Tips
 TI-83+ can calculate
a confidence interval
for a population
proportion.
 STAT
 TESTS
 A: 1-PROPZInt
TI-83+ Tips
 Enter the number of
successes observed
and the sample size.
 Specify a confidence
level and then
Calculate.
Caution! Caution! Caution!
Don’t mistake what the interval means:
Do not suggest that the parameter varies.
The population parameter is fixed; the interval varies
from sample to sample.
Do not claim that other samples will agree with
this sample.
The interval isn’t about sample proportions; it is about
the population proportion.
Don’t be certain about the parameter.
We can’t be absolutely certain that the population
proportion isn’t outside the interval – just pretty sure.
Caution! Caution! Caution!
Don’t forget: it’s a parameter.
The confidence interval is about the unknown
population parameter, p.
Don’t claim too much.
Write your confidence statement about your sample.
Take responsibility.
Confidence intervals are about uncertainty. You are
uncertain, however, not the parameter.
Margin of Error: Too Large to be Useful?
Think about the margin of error during
design of the study.
Choose a larger sample to reduce
variability in the sample proportion.
To cut the standard error (and the ME) in
half, quadruple the sample size.
Remember, though, that bigger samples
cost more money and effort.
Margin of Error: An Example
 Suppose a candidate is planning a poll and
wants to estimate voter support within 3% with
95% confidence. How large a sample is
needed?
pq
pq
0.03  1.96
n
n
Worst case (largest sample size): p  .5
ME  z *
0.03  1.96
n
1.96
.5 .5 
n
.5 .5 
0.03 n  1.96
 32.67
.5 .5 
n   32.67   1067.1
2
0.03
Round up, so sample size needs to be 1068 to keep the margin of error
as small as 3% with a confidence level of 95%.
Violation of Assumptions
Watch out for biased samples.
Check potential sources of bias.
Relying on voluntary response
Undercoverage of the population
Nonresponse bias
Response bias
Think about independence.