Stata Calculate Ci Given N and Proportion

Calculating confidence intervals in Stata when you have a sample size (n) and proportion is a common statistical task. This guide explains how to perform the calculation, interpret the results, and use the interactive calculator provided.

What is a Confidence Interval?

A confidence interval (CI) is a range of values that is likely to contain the true population parameter with a certain level of confidence. For proportions, the confidence interval estimates the range within which the true population proportion is likely to fall.

Common confidence levels are 90%, 95%, and 99%. The most commonly used is 95%, which means we are 95% confident that the true population proportion falls within the calculated interval.

How to Calculate CI in Stata

In Stata, you can calculate a confidence interval for a proportion using the ci command. The formula for the confidence interval is:

Confidence Interval Formula:

CI = p ± z*√(p*(1-p)/n)

Where:

p = sample proportion
z = z-score corresponding to the desired confidence level
n = sample size

The z-score for common confidence levels are:

90% confidence: z = 1.645
95% confidence: z = 1.960
99% confidence: z = 2.576

Assumptions:

The sample is randomly selected from the population
The sample size is large enough (n ≥ 30 is often recommended)
There are at least 5 successes and 5 failures in the sample

Worked Example

Let's calculate a 95% confidence interval for a proportion where:

Sample size (n) = 100
Sample proportion (p) = 0.60 (60%)

Using the formula:

CI = 0.60 ± 1.960*√(0.60*(1-0.60)/100)

= 0.60 ± 1.960*√(0.24/100)

= 0.60 ± 1.960*0.04899

= 0.60 ± 0.0960

Lower bound = 0.60 - 0.0960 = 0.504

Upper bound = 0.60 + 0.0960 = 0.696

The 95% confidence interval is (0.504, 0.696). This means we are 95% confident that the true population proportion falls between 50.4% and 69.6%.

Interpreting Results

When interpreting confidence intervals for proportions:

If the interval includes 0.5 (50%), the result is not statistically significant
If the interval does not include 0.5, the result is statistically significant
Wider intervals indicate less precision in the estimate
Narrower intervals indicate more precise estimates

For example, if your confidence interval is (0.45, 0.55), it includes 0.5, suggesting the proportion is not significantly different from 50%. If your interval is (0.60, 0.70), it does not include 0.5, suggesting the proportion is significantly greater than 50%.

FAQ

What is the difference between a confidence interval and a margin of error?

A confidence interval is a range of values that is likely to contain the true population parameter. The margin of error is half the width of the confidence interval. For example, if the confidence interval is (0.50, 0.70), the margin of error is 0.10.

How do I know if my sample size is large enough?

For proportions, a common rule of thumb is that the sample size should be at least 30, and there should be at least 5 successes and 5 failures in the sample. Larger sample sizes provide more precise estimates.

What does a 95% confidence level mean?

A 95% confidence level means that if you were to take 100 different samples and calculate 95% confidence intervals for each, you would expect approximately 95 of those intervals to contain the true population proportion.