Interval
Estimation for Proportions
(Margin of Error)
Interval
estimation
(margin of error) uses sample data to determine a range (interval) that, at an established
level of confidence, will contain the population proportion.
Steps
Determine
the confidence level (alpha
is generally .05)
Use
the z-distribution
table to find the critical value for a 2-tailed test given the selected confidence level
(alpha)
Estimate
the standard error of the proportion
where
p
= sample proportion
q=1-p
Estimate
the confidence interval
CV
= critical value
CI
= p ± (CV)(Sp)
Interpret
Based
on alpha
.05, you are 95% confident that the proportion in the population from which the sample was
obtained is between __ and __.
Note:
Given the sample data and level of error, the confidence interval provides an estimated
range of proportions that is most likely to contain the population proportion. The term
"most likely" is measured by alpha (i.e., in most cases there is a 5% chance --alpha .05-- that the
confidence interval does not contain the true population proportion).
More
About the Standard Error of the Proportion
The
standard error of the proportion will vary as sample size and the proportion changes. As
the standard error increases, so will the margin of error.
|
Sample
Size (n) |
Proportion
(p) |
100 |
300 |
500 |
1000 |
5000 |
10000 |
0.9 |
0.030 |
0.017 |
0.013 |
0.009 |
0.004 |
0.003 |
0.8 |
0.040 |
0.023 |
0.018 |
0.013 |
0.006 |
0.004 |
0.7 |
0.046 |
0.026 |
0.020 |
0.014 |
0.006 |
0.005 |
0.6 |
0.049 |
0.028 |
0.022 |
0.015 |
0.007 |
0.005 |
0.5 |
0.050 |
0.029 |
0.022 |
0.016 |
0.007 |
0.005 |
0.4 |
0.049 |
0.028 |
0.022 |
0.015 |
0.007 |
0.005 |
0.3 |
0.046 |
0.026 |
0.020 |
0.014 |
0.006 |
0.005 |
0.2 |
0.040 |
0.023 |
0.018 |
0.013 |
0.006 |
0.004 |
0.1 |
0.030 |
0.017 |
0.013 |
0.009 |
0.004 |
0.003 |
Effect
of changes in the proportion
As
a proportion approaches .5 the error will be at its greatest value for a given sample
size. Proportions close to 0 or 1 will have
the lowest error.
The
error above a proportion of .5 is a mirror reflection of the error below a proportion of
.5.
Effect
of changes in sample size
As
sample size increases the error of the proportion will decrease for a given proportion.
The
reduction in error of the proportion as sample size increases is not constant. As an example, at a proportion of 0.9, increasing
the sample size from 100 to 300 cut the standard error by about half (from .03 to .017). Increasing the sample size by another 200 only
reduced the standard error by about one quarter (.017 to .013).
Example: Interval
Estimation for Proportions
Problem:
A random sample of 500 employed adults found that 23% had traveled to a foreign country.
Based on these data, what is your estimate for the entire employed adult population?
n=500,
p = .23 q = .77
Use
alpha
.05 (i.e., the critical value is 1.96)
Estimate
Sampling Error
Compute
Interval
Interpret
You
are 95% confident that the actual proportion of all employed adults who have traveled to a
foreign country is between 19.3% and 26.7%.
Software
Output Example
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