Confidence Interval Builder

Formula

CI = x̄ ± z × (σ/√n)

The confidence interval equals the sample mean plus/minus the z-score times the standard error. Standard error is population standard deviation divided by square root of sample size.

Worked Examples

Example 1: Survey Satisfaction Score

Problem:A survey of 200 customers shows mean satisfaction of 7.5/10 with SD=1.8. Calculate 95% CI.

Solution:Given: x̄=7.5, n=200, σ=1.8, z=1.96 SE = 1.8/√200 = 0.127 MoE = 1.96 × 0.127 = 0.249 95% CI = 7.5 ± 0.249 = (7.251, 7.749) Interpretation: We're 95% confident true mean satisfaction is between 7.25 and 7.75.

Result:95% CI: (7.25, 7.75) | MoE = 0.25 points

Example 2: Election Poll

Problem:Poll of 1,000 voters shows 52% support candidate A. Calculate 95% CI for proportion.

Solution:Given: p̂=0.52, n=1000, z=1.96 SE = √(0.52×0.48/1000) = 0.0158 MoE = 1.96 × 0.0158 = 0.031 95% CI = 0.52 ± 0.031 = (0.489, 0.551) or 48.9% to 55.1% Since CI includes 50%, result is not statistically conclusive.

Result:95% CI: (48.9%, 55.1%) | Too close to call

Example 3: Manufacturing Quality

Problem:Sample of 50 parts has mean length 10.02mm, SD=0.05mm. Find 99% CI.

Solution:Given: x̄=10.02, n=50, σ=0.05, z=2.576 SE = 0.05/√50 = 0.00707 MoE = 2.576 × 0.00707 = 0.0182 99% CI = 10.02 ± 0.0182 = (10.0018, 10.0382) Narrower than typical due to low variance.

Result:99% CI: (10.002, 10.038) | Precision ±0.018mm

Frequently Asked Questions

What is a confidence interval?

A confidence interval is a range of values that likely contains the true population parameter. A 95% CI means if we repeated the sampling process many times, 95% of the calculated intervals would contain the true value. It quantifies uncertainty in our estimate.

What does 95% confidence level mean?

95% confidence means that if you were to take 100 different samples and compute a CI for each, approximately 95 of those intervals would contain the true population parameter. It does NOT mean there's a 95% probability the true value is in this specific interval.

How does sample size affect CI width?

Larger samples produce narrower confidence intervals. The margin of error decreases proportionally to the square root of n. To halve the CI width, you need 4x the sample size. This is why increasing from n=100 to n=400 has the same effect as n=25 to n=100.

What is the margin of error?

Margin of error (MoE) is the half-width of the confidence interval: MoE = z × (σ/√n). It represents the maximum expected difference between the sample statistic and true population parameter at the given confidence level.

How do I interpret overlapping confidence intervals?

Overlapping CIs suggest the difference may not be statistically significant, but this is not a definitive test. Two groups with overlapping 95% CIs can still have p < 0.05 difference. Use formal hypothesis tests for comparison.

What affects confidence interval width?

CI width is determined by: 1) Confidence level (higher = wider), 2) Sample size (larger = narrower), 3) Population variability (more variable = wider), 4) Distribution (t wider than z for small n).

Background & Theory

Confidence intervals provide a range estimate for population parameters, incorporating sample variability. Unlike point estimates, CIs communicate precision. The interval is constructed using the sampling distribution of the statistic, typically normal for large samples (Central Limit Theorem) or t-distribution for small samples with unknown population variance.

History

Confidence intervals were introduced by Jerzy Neyman in 1937, building on earlier work by Ronald Fisher on estimation. Neyman's frequentist interpretation became the standard: the interval either contains the parameter or not, but the procedure produces correct intervals 95% of the time. The concept formalized the idea of quantifying uncertainty in statistical estimates.