Confidence Interval for Proportion
Learn how to estimate an unknown population proportion using a sample proportion and a confidence interval.
What Are We Estimating?
Sometimes we are interested in a percentage or proportion rather than an average.
For example, a company may want to estimate the percentage of all customers who are satisfied with its service.
We can survey a sample, calculate the sample proportion, and use it to estimate the unknown population proportion.
Population
Unknown proportion p
Sample
Sample proportion p̂
Confidence Interval
Range for p
What You Will Learn
Population proportion vs sample proportion
How to calculate a sample proportion
Standard error for a proportion
Confidence interval formula for a proportion
Margin of error
Conditions for the normal approximation
How sample size affects precision
Real-world analytics applications
Population Proportion vs Sample Proportion
Population Proportion
The proportion of the entire population having a particular characteristic.
Sample Proportion
The proportion calculated from the observations in a sample.
Key idea:
We usually do not know p. We calculate p̂ from the sample and use it to estimate p.
Calculate the Sample Proportion
Suppose an online store surveys 200 customers and 160 customers say they are satisfied.
Number satisfied
160
Sample size
200
p̂ = 160 / 200
p̂ = 0.80 = 80%
The sample proportion of 80% is our point estimate of the population proportion.
Standard Error of a Proportion
For the common large-sample normal approximation, the estimated standard error of a sample proportion is:
SE = √[p̂(1 − p̂) / n]
Where:
- p̂ = sample proportion
- n = sample size
- 1 − p̂ = proportion not having the characteristic
Confidence Interval Formula
Under the usual large-sample normal approximation, a confidence interval for a population proportion can be written as:
p̂ ± z* × √[p̂(1 − p̂) / n]
p̂
Point estimate
z*
Critical value
SE
Estimated standard error
Worked Example
An online store surveys 200 customers. 160 customers report that they are satisfied.
Sample size
n = 200
Successes
160
Sample proportion
0.80
1. Calculate Standard Error
SE = √[0.80 × 0.20 / 200]
SE = 0.0283 approximately
2. Use the 95% Critical Value
z* = 1.96
3. Calculate Margin of Error
ME = 1.96 × 0.0283 ≈ 0.0554
4. Construct the Interval
0.80 ± 0.0554
Approximately 74.46% to 85.54%
Conditions for the Normal Approximation
The simple normal-approximation interval works best when the sample contains enough observations in both outcome categories.
Expected successes
n p̂
Should generally be sufficiently large.
Expected failures
n(1 − p̂)
Should also generally be sufficiently large.
Practical note:
Textbooks often use rules such as at least 10 expected successes and 10 expected failures for the basic normal approximation. For small samples or extreme proportions, more accurate methods such as Wilson or exact methods may be preferable.
How Sample Size Affects Precision
Increasing the sample size generally reduces the standard error of a sample proportion and can make the confidence interval narrower.
Smaller Sample
More uncertainty
Larger Sample
Less uncertainty
More Information
Greater precision
Visualizing a Proportion
Suppose 80% of a sample of customers are satisfied.
The sample proportion gives us an estimate, but sampling variation means another random sample could produce a slightly different proportion.
Interpreting the Confidence Interval
Example:
95% CI ≈ 74.46% to 85.54%
The confidence interval procedure is designed so that, over many repeated random samples under the same conditions, about 95% of the resulting intervals would contain the true population proportion.
Common Mistake
A 95% confidence interval does not mean that 95% of the population lies inside the interval. It is estimating a population proportion, which is a single fixed parameter.
Customer Conversion Analysis
A digital marketing team wants to estimate the percentage of website visitors who make a purchase.
Visitors
1,000
Purchases
120
Sample proportion
12%
Confidence
95%
The analytics team can construct a confidence interval around the 12% sample conversion rate.
This provides a range of plausible values for the population conversion proportion instead of treating 12% as if it were known exactly.
Test Your Understanding
CHECK YOUR UNDERSTANDING
What population quantity is estimated by a confidence interval for a proportion?
CHECK YOUR UNDERSTANDING
A sample has 160 satisfied customers out of 200. What is the sample proportion?
CHECK YOUR UNDERSTANDING
Which expression represents the estimated standard error of a sample proportion?
CHECK YOUR UNDERSTANDING
If the sample size increases while the other relevant factors remain similar, what generally happens to the confidence interval?
CHECK YOUR UNDERSTANDING
For the basic normal approximation, what should generally be sufficiently large?
CHECK YOUR UNDERSTANDING
A sample proportion is 0.80 and the margin of error is 0.055. What is the approximate interval?
Fill in the Blanks
The sample proportion is commonly represented by p____.
A confidence interval for a proportion estimates the population ______.
The standard error of a sample proportion decreases as the sample size generally gets ______.
The interval is constructed around the sample ______.
App Satisfaction Survey
A company surveys 400 users. 320 users say they are satisfied with the app.
Step 1: Calculate the sample proportion.
p̂ = 320 / 400 = 0.80
Step 2: Calculate the standard error.
SE = √(0.80 × 0.20 / 400)
SE ≈ 0.02
Step 3: Using z* = 1.96, calculate the margin of error.
ME = 1.96 × 0.02 = 0.0392
Approximate 95% CI = 76.08% to 83.92%
Final Challenge
Two surveys estimate the same population proportion at the same confidence level.
Survey A
n = 100
Margin of error = 0.10
Survey B
n = 400
Margin of error = 0.05
What does this demonstrate?
With other relevant factors held constant, the larger sample can produce a smaller standard error and a narrower confidence interval.
Lesson Summary
A confidence interval for a proportion estimates the unknown population proportion p.
The sample proportion p̂ is the point estimate.
The estimated standard error is √[p̂(1 − p̂) / n] for the common normal approximation.
The confidence interval has the basic form p̂ ± critical value × standard error.
The normal approximation requires sufficiently large expected successes and failures.
Larger samples generally reduce standard error and improve precision.
A confidence interval communicates uncertainty caused by sampling variation.
For small samples or extreme proportions, more accurate interval methods may be preferable.