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Confidence Interval Formulas
Confidence Interval
Formula Breakdown
Interpretation Guide
Correct interpretation: If we repeated this sampling process many times, approximately 95% of the resulting confidence intervals would contain the true population mean.
Note: This does not mean there is a 95% probability the true value lies in this specific interval.
| Interval Type | Use When | Critical Value |
|---|---|---|
| z-interval | σ is known | z* from N(0,1) |
| t-interval | σ unknown, use s | t* from t(n-1) |
| Proportion (Wald) | np̂ ≥ 5, n(1-p̂) ≥ 5 | z* from N(0,1) |
Model Assumptions
- Sample is a simple random sample from the population
- Observations are independent
- For t-interval: population is approximately normal, or n is large (CLT)
For educational purposes. Not financial advice. Market conventions simplified.
Understanding Confidence Intervals
What is a Confidence Interval?
A confidence interval is a range of values constructed from sample data that is designed to contain the true population parameter with a specified probability over repeated sampling. Unlike a point estimate (single value), a confidence interval quantifies the uncertainty in our estimate.
The margin of error determines interval width
When to Use Each Interval Type
Z-Interval
Known σ
Use when population standard deviation is truly known (rare in practice). Formula: x̄ ± z* × (σ/√n)
T-Interval
Unknown σ
Use when estimating σ from sample. Accounts for extra uncertainty with wider intervals. Formula: x̄ ± t* × (s/√n)
Common Misinterpretation
Correct: "If we repeated this sampling process many times, approximately 95% of the resulting confidence intervals would contain the true population parameter."
The true parameter is fixed (not random). The randomness comes from the sampling process, not the parameter itself.
Key Assumptions
- Random sampling: Sample must be randomly selected from the population
- Independence: Observations must be independent of each other
- For t-interval: Population approximately normal OR large sample (n ≥ 30 by rule of thumb)
- For proportion interval: np̂ ≥ 5 and n(1-p̂) ≥ 5 for normal approximation validity
Frequently Asked Questions
Disclaimer
This calculator is for educational purposes only. It uses standard statistical formulas and assumptions. The t-interval assumes approximately normal populations for small samples. The Wald proportion interval uses normal approximation and may produce bounds outside [0, 1] for extreme proportions. For critical applications, verify results with statistical software and consider alternative interval methods. This tool should not be used as the sole basis for important decisions.
