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Applied Statistics Handbook
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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


Google

 

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