Confidence Interval Calculator

Calculate confidence intervals for means and proportions. Find the range of values likely to contain the true population parameter.

Confidence Interval

95% CI

71.6738 - 78.3262

±3.3262

Point Estimate75.0000
Margin of Error3.3262
Critical Value1.9600

Confidence Interval Type

Input Values

95% Confidence Interval

We are 95% confident the true mean is between:

71.6738 — 78.3262

Point Estimate

75.0000

Margin of Error

±3.3262

Standard Error

1.6971

Critical Value

1.9600

Visual Representation

71.67
75.00
78.33
95% Confidence IntervalPoint Estimate

Calculation Steps

x̄ ± t(α/2) × (s/√n)

1. Critical value (t): 1.9600

2. Standard Error: 1.697056

3. Margin of Error: 1.9600 × 1.697056 = 3.326230

4. CI: 75.0000 ± 3.3262

Critical Values Reference

Confidence Levelαα/2z-value
80%0.200.101.282
90%0.100.051.645
95%0.050.0251.960
99%0.010.0052.576
99.9%0.0010.00053.291

?How Do You Calculate a Confidence Interval?

A confidence interval estimates a population parameter with a margin of error. For a sample mean: CI = x-bar +/- (z * s/sqrt(n)). At 95% confidence, z=1.96. If sample mean is 100, SD is 15, n=30: CI = 100 +/- (1.96 * 15/5.48) = 100 +/- 5.37, giving interval [94.63, 105.37].

What is a Confidence Interval?

A confidence interval is a range of values that likely contains an unknown population parameter. It provides both an estimate and a measure of uncertainty. The confidence level (e.g., 95%) indicates how often such intervals would contain the true parameter if the study were repeated many times.

Key Facts About Confidence Intervals

  • Confidence interval: estimate +/- margin of error
  • Common confidence levels: 90%, 95%, 99%
  • Higher confidence = wider interval
  • Z-scores: 90%=1.645, 95%=1.96, 99%=2.576
  • CI for mean: x-bar +/- z*(s/sqrt(n))
  • CI for proportion: p +/- z*sqrt(p(1-p)/n)
  • Larger sample size = narrower interval
  • 95% CI: we are 95% confident the true value is within this range

Quick Answer

A confidence interval estimates a population parameter with a margin of error. For a sample mean: CI = x-bar +/- (z * s/sqrt(n)). At 95% confidence, z=1.96. If sample mean is 100, SD is 15, n=30: CI = 100 +/- (1.96 * 15/5.48) = 100 +/- 5.37, giving interval [94.63, 105.37].

Frequently Asked Questions

A confidence interval is a range of values that likely contains the true population parameter. A 95% CI means if we repeated the sampling many times, about 95% of intervals would contain the true value. It expresses uncertainty in our estimate.
The confidence level (e.g., 95%) indicates how often the interval would contain the true parameter if we repeated the study many times. It's NOT the probability that the true value is in THIS interval (it either is or isn't).
Use t-distribution when: (1) sample size is small (n < 30), (2) population standard deviation is unknown, (3) estimating a mean. Use z-distribution for: proportions, large samples, or when population σ is known.
To reduce CI width: (1) increase sample size (most effective), (2) lower confidence level (90% instead of 95%), (3) reduce variability in data (if possible through better measurement). Trade-offs exist between width and confidence.
Margin of error is half the width of the confidence interval. It represents the maximum expected difference between the sample estimate and true population value at a given confidence level. Often reported as ±E in surveys.

Last updated: 2025-01-15