Confidence Interval
95% CI
71.5896 - 78.4104
±3.4104
Confidence Interval Type
Input Values
95% Confidence Interval
We are 95% confident the true mean is between:
71.5896 — 78.4104
Point Estimate
75.0000
Margin of Error
±3.4104
Standard Error
1.6971
Critical Value
2.0096
Visual Representation
Calculation Steps
x̄ ± t(α/2) × (s/√n)
1. Critical value (t): 2.0096
2. Standard Error: 1.697056
3. Margin of Error: 2.0096 × 1.697056 = 3.410362
4. CI: 75.0000 ± 3.4104
Critical Values Reference
| Confidence Level | α | α/2 | z-value |
|---|---|---|---|
| 80% | 0.20 | 0.10 | 1.282 |
| 90% | 0.10 | 0.05 | 1.645 |
| 95% | 0.05 | 0.025 | 1.960 |
| 99% | 0.01 | 0.005 | 2.576 |
| 99.9% | 0.001 | 0.0005 | 3.291 |
?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
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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
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 many times, about 95% of intervals would contain the true value. It expresses uncertainty in our estimate.
What does the confidence level mean?
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).
When should I use t-distribution vs z-distribution?
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.
How can I make my confidence interval narrower?
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.
What is margin of error?
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: 2026-01-15
How this works
Formulas follow standard definitions from the NIST Digital Library of Mathematical Functions and classical textbook derivations. Calculations run entirely in your browser. Where a closed-form solution exists, it is used; where an iterative or numerical method is required, the implementation is named on the page.
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
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