Confidence Intervals for Proportions
In the previous lesson, we learned how to build confidence intervals for means. In many real-world studies, however, we are interested in percentages or proportions rather than averages.
This lesson explains how to construct and interpret confidence intervals for population proportions.
What Is a Population Proportion?
A population proportion represents the fraction or percentage of a population that has a certain characteristic.
Examples include:
- Percentage of voters supporting a candidate
- Proportion of defective products
- Share of customers satisfied with a service
Sample Proportion as a Point Estimate
The sample proportion, usually denoted by p̂, is used as a point estimate of the population proportion.
It is calculated as:
p̂ = Number of successes ÷ Sample size
Numerical Example
In a survey of 200 customers, 120 say they are satisfied.
The sample proportion is:
p̂ = 120 ÷ 200 = 0.60
So, 60% is our point estimate of the population proportion.
Confidence Interval for a Proportion
A confidence interval for a proportion follows the same idea as for means:
Confidence Interval = Sample Proportion ± Margin of Error
The margin of error depends on:
- Sample size
- Confidence level
- Variability in the data
Simple Example of a Confidence Interval
Suppose a poll reports that 52% of voters support a policy with a margin of error of ±4%.
The confidence interval is:
(48%, 56%)
This means the true population support likely lies between 48% and 56%.
When Confidence Intervals for Proportions Are Valid
To use confidence intervals for proportions reliably:
- The sample should be random
- The sample size should be large enough
- There should be enough successes and failures
These conditions help ensure the sampling distribution is approximately normal.
Effect of Sample Size
Just like confidence intervals for means:
- Larger samples produce narrower intervals
- Smaller samples produce wider intervals
Increasing sample size improves precision.
Real-World Example
An online platform surveys 1,000 users and finds that 72% prefer a new feature.
If the margin of error is ±2%, the confidence interval becomes:
(70%, 74%)
This gives decision-makers a realistic range instead of a single number.
Common Misinterpretations
- The interval does not guarantee the true proportion is inside
- Confidence refers to the method, not the specific interval
- Bias in sampling affects the interval
Quick Check
If a confidence interval is (0.45, 0.55), what does this represent?
Practice Quiz
Question 1:
What statistic is used as a point estimate for a population proportion?
Question 2:
What happens to the confidence interval if sample size increases?
Question 3:
Does a confidence interval fix biased survey data?
Mini Practice
A survey of 500 people finds that 275 prefer Product A.
- What is the sample proportion?
- If the margin of error is ±5%, what is the confidence interval?
What’s Next
In the next lesson, we will begin Hypothesis Testing, starting with its core ideas and terminology.