Confidence Interval Calculator | Normal Population Mean Estimation

Input Parameters

Enter your sample data and select the appropriate scenario

Sample Mean (

\bar{x}

)

Sample Size (n)

Population Variance Status

Sample Standard Deviation (s)

Confidence Level (1 - α)

Pivotal Quantity Method

Known σ²: Use pivotal quantity $G = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1)$

Unknown σ²: Use pivotal quantity $T = \frac{\bar{X} - \mu}{S/\sqrt{n}} \sim t(n-1)$

Principle: Find constants c, d such that $P\{c \leq G \leq d\} = 1-\alpha$

Result: Rearrange inequality to get $P\{\hat{\theta}_L \leq \mu \leq \hat{\theta}_U\} = 1-\alpha$

Please fill in all required fields with valid positive numbers to calculate the confidence interval.

📚 Confidence Interval Theory

Understanding the mathematical foundations and practical interpretations

Coverage Probability

The coverage probability measures how often confidence intervals contain the true parameter:

P_\mu\{\mu \in [\hat{\theta}_L, \hat{\theta}_U]\}

Key insight: The parameter μ is fixed, but the interval is random (depends on sample data).

Confidence coefficient: Minimum coverage probability over all possible μ values.

Distribution Selection

Known σ²: Use standard normal distribution

\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \sim N(0,1)

Unknown σ²: Use t-distribution with n-1 degrees of freedom

\frac{\bar{X} - \mu}{S/\sqrt{n}} \sim t(n-1)

Large n: t-distribution approaches standard normal (n ≥ 30 approximation).

Interval Properties

Length vs Confidence: Higher confidence → wider intervals

Sample size effect: Larger n → narrower intervals

Precision measure: Expected interval length

E[\hat{\theta}_U - \hat{\theta}_L]

Neyman principle: Minimize expected length subject to confidence level constraint.

Practical Considerations

Assumption checking: Normality, independence, random sampling

Robustness: t-intervals robust to mild non-normality (large n)

One-sided intervals: Upper/lower bounds with confidence level 1-α

Bootstrap alternative: For non-normal populations or small samples

Reporting: Always state confidence level and sample assumptions