🎓 Statistics & Data • Inferential Statistics
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Confidence Interval Calculator
Computes lower and upper bounds of confidence intervals x̄ ± z*·(s/√n) for population parameters.
Presets:
Standard Deviation Known?
95% Two-Sided Confidence Interval
[71.856, 78.944]
Margin of Error±3.544
Standard Error (SE)1.7590
Critical Value2.015t* (df=44)
Total Span (Width)7.089
⚡
Benchmark Value (μ₀ = 70): Falls outside the 95% interval [71.856, 78.944]. Reject H₀ at α = 0.05. Statistically significant difference from benchmark.
Confidence Interval Visual RangeMean/Estimate Center ± 3.544
Formula & Derivation
Standard Error: SE = 1.7590
Critical Value: 2.015
Margin of Error: MOE = critical × SE = ±3.544
Interval = estimate ± MOE = [71.856, 78.944]
What is a Confidence Interval?
A confidence interval is an estimated range of values likely to contain the true population parameter at a specified confidence level (e.g. 95%).
Formula & Step-by-Step Calculation
CI = x̄ ± z* · (s / √n) (or p̂ ± z* · √[p̂(1-p̂)/n])
Point estimate plus or minus the margin of error.
Worked Step-by-Step Examples
Example 1
Sample mean 50, std dev 10, n = 100 at 95% confidence level
Solution: CI = [48.04, 51.96] (Margin of Error = ±1.96)
• SE = 10 / √100 = 1.0; MOE = 1.96 × 1.0 = 1.96 ⟹ [50 - 1.96, 50 + 1.96]
Common Real-World & Academic Use Cases
- ✓ Medical clinical trial biomarker parameter bounds
- ✓ Political election polling percentage bands
- ✓ Financial portfolio risk interval forecasting
How to Use the Confidence Interval Calculator
1
Enter Sample Mean & SD
Input x̄ and s.
2
Select Confidence Level
Choose 90%, 95%, or 99%.
3
Read Interval Bounds
Inspect lower and upper confidence limits.
Frequently Asked Questions
Q: What does 95% confidence actually mean?
If we repeated the experiment 100 times, 95 of the constructed confidence intervals would contain the true population mean.