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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:
Because the sampling distribution model is Normal: 68% of all samples will be within
95% of all samples will be within
99.5% of all samples will be within
pq
SE pn
1p SE
2p SE
3p SE
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
is called the margin of error (ME).
In general, confidence intervals are written as:
There is a conflict between certainty and
precision Choose a confidence level – the data does not
determine the confidence level
p
estimate ME
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
.
* *
, the confidence interval is
. The critical value, , depends
on the particular confidence level, C, that you specify
pqSE p
n
p z SE p z
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?
*
2
0.03 1.96
Worst case (largest sample size): .5
.5 .50.03 1.96 0.03 1.96 .5 .5
1.96 .5 .532.67 32.67 1067.1
0.03Round up, so sample size needs to be 1068 to keep the margin of er
pq pqME z
n np
nn
n n
ror
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.
