Confidence Interval Calculator Step by Step
Calculate confidence intervals with complete step-by-step explanations. Learn statistical confidence, margin of error, and hypothesis testing. Free calculator.
Quick Answer
Confidence interval: CI = x̄ ± z(σ/√n) for known σ, or x̄ ± t(s/√n) for unknown σ. Range where true population parameter likely falls. 95% CI means 95% confidence true value lies within range. Essential for statistical inference and decision-making.
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Calculate With Full ToolWhat is Confidence Interval?
A confidence interval is a range of values that likely contains the true population parameter with a specified level of confidence. It provides a measure of uncertainty around sample estimates and is fundamental to statistical inference and hypothesis testing.
How Confidence Intervals Work
Confidence intervals are calculated using sample statistics, critical values from probability distributions, and the standard error. The width depends on sample size, variability, and confidence level. Larger samples and lower confidence levels produce narrower intervals.
Interpreting Confidence Levels
A 95% confidence level means that if we repeated the sampling process many times, 95% of the calculated intervals would contain the true population parameter. It does not mean there's a 95% probability the parameter is in this specific interval.
Confidence Interval Formulas
Known σ: CI = x̄ ± z(σ/√n)
Unknown σ: CI = x̄ ± t(s/√n)
Proportion: CI = p̂ ± z√(p̂(1-p̂)/n)
Known Population SD: CI = x̄ ± z(σ/√n)
Unknown Population SD: CI = x̄ ± t(s/√n)
Proportion: CI = p̂ ± z√(p̂(1-p̂)/n)
Critical Values: z₁.₉₆ = 1.96 (95%), z₂.₅₈ = 2.58 (99%)
Margin of Error: ME = critical value × standard error
Step-by-Step Example
Example: Sample mean = 85, s = 12, n = 36, 95% confidence
Step 1: Identify parameters: x̄ = 85, s = 12, n = 36, confidence = 95%
Step 2: Find critical value: t₀.₀₂₅,₃₅ ≈ 2.03 (from t-table)
Step 3: Calculate standard error: s/√n = 12/√36 = 12/6 = 2
Step 4: Calculate margin of error: ME = 2.03 × 2 = 4.06
Step 5: Calculate CI: 85 ± 4.06 = (80.94, 89.06)
Step 6: Interpret: 95% confident true mean is between 80.94 and 89.06
Example: Proportion with p̂ = 0.6, n = 100, 95% confidence
Step 1: Identify parameters: p̂ = 0.6, n = 100, confidence = 95%
Step 2: Find critical value: z₁.₉₆ = 1.96
Step 3: Calculate standard error: √(0.6×0.4/100) = √0.0024 = 0.049
Step 4: Calculate margin of error: ME = 1.96 × 0.049 = 0.096
Step 5: Calculate CI: 0.6 ± 0.096 = (0.504, 0.696)
Step 6: Interpret: 95% confident true proportion is between 50.4% and 69.6%
These examples demonstrate confidence interval calculation for both means and proportions. The intervals provide ranges where we're confident the true population parameters lie.
Who Should Use This Calculator?
Researchers
Estimate population parameters from samples
Statisticians
Perform hypothesis testing and inference
Quality Control
Monitor process parameters and tolerances
Pollsters
Calculate margins of error in surveys
Frequently Asked Questions
What's the difference between 95% and 99% confidence intervals?
A 99% confidence interval is wider than a 95% interval because it requires more certainty. Higher confidence levels provide greater assurance but less precision, while lower levels are more precise but less certain.
When should I use t-distribution vs z-distribution?
Use t-distribution when population standard deviation is unknown and sample size is small (n < 30). Use z-distribution when population standard deviation is known or sample size is large (n ≥ 30) due to the Central Limit Theorem.
What does it mean if confidence intervals overlap?
Overlapping confidence intervals suggest that the difference between groups may not be statistically significant. However, formal hypothesis testing is needed to determine significance, as visual overlap alone is not conclusive.
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