Confidence Interval Calculator
Calculate confidence intervals for population means and proportions with Z and t-distributions
Confidence Interval for Population Mean
A confidence interval for a population mean estimates the range of values within which the true population mean is likely to fall. The interval depends on the sample size, confidence level, and whether the population standard deviation is known.
Formulas
90% Confidence
90% of intervals will contain the true population mean
95% Confidence
Most commonly used confidence level in research
99% Confidence
Higher confidence but wider interval
Confidence Interval for Population Proportion
A confidence interval for a population proportion estimates the range of values within which the true population proportion is likely to fall. This is commonly used in surveys, polls, and quality control.
Formula
Sample Requirements
np̂ ≥ 5 and n(1-p̂) ≥ 5 for normal approximation
Applications
Polls, surveys, quality control, medical studies
Confidence Interval for Population Mean
Sample Data
Example: Heights of 25 students (inches)
Sample Size: 25
Sample Std Dev: 3.2 inches
Example: Survey of 200 voters
Sample Size: 200
Sample Proportion: 0.60 (60%)
Results
📊 Detailed Analysis
Confidence Interval Visualization
Confidence Interval Guide
Understanding Confidence Intervals
Confidence Levels
90% Confidence
If we repeated this study 100 times, about 90 of the intervals would contain the true population parameter.
95% Confidence
Most commonly used. About 95 out of 100 intervals would contain the true parameter.
99% Confidence
Higher confidence but wider intervals. About 99 out of 100 intervals would contain the true parameter.
Distribution Choice
Z-Distribution
Use when population standard deviation (σ) is known or sample size is large (n ≥ 30).
t-Distribution
Use when population standard deviation is unknown and sample size is small (n < 30).
Real-World Applications
Population Means
- • Average height/weight in populations
- • Mean test scores
- • Average income levels
- • Manufacturing quality control
Population Proportions
- • Election polling results
- • Market research surveys
- • Medical treatment success rates
- • Product defect rates
Business Applications
- • Customer satisfaction rates
- • Sales forecasting ranges
- • Risk assessment
- • A/B testing results
Key Concepts & Assumptions
Important Concepts
- • Margin of Error: Half the width of the confidence interval
- • Critical Value: The z or t value that corresponds to your confidence level
- • Standard Error: Measures the precision of your sample statistic
- • Degrees of Freedom: n-1 for t-distribution calculations
Assumptions & Limitations
- • Random Sampling: Sample must be randomly selected from the population
- • Independence: Sample observations must be independent
- • Normality: Population should be approximately normal (or large sample size)
- • Sample Size: For proportions, np̂ ≥ 5 and n(1-p̂) ≥ 5
Interpretation Tips
What Confidence Intervals Tell Us
✅ Correct: "We are 95% confident that the true population mean lies between X and Y."
✅ Correct: "If we repeated this study many times, 95% of our intervals would contain the true parameter."
❌ Incorrect: "There is a 95% probability that the true mean is in this interval."
❌ Incorrect: "95% of the data falls within this interval."