Lecture Notes: Confidence Interval

Introduction

• Confidence Interval (CI): A range of values used to estimate an unknown statistical parameter.
• Purpose: Provides a range instead of a single point estimate.
• Example: Instead of saying "average screen time is 3 hours", a CI might suggest 2 to 4 hours with a 95% confidence level.

History

• Early Developments: Methods for binomial proportion CI date back to the 1920s.
• Key Figure: Jerzy Neyman presented a thorough account of CI in 1937.
• Medical Use: CI gained prominence in medical journals in the 1970s and became a requirement by 1988.

Definition

• Mathematical Definition: CI for a parameter ( \theta ) is defined by random variables ( u(X) ) and ( v(X) ), with confidence level ( \gamma ).
• Confidence Level: Typically close to 1 (e.g., 95%) indicating reliability in repeated sampling.
• Approximation: Exact confidence levels are hard to achieve; approximations are often used.

Methods of Derivation

• Bootstrapping: A widely applicable method for CI calculation.
• Central Limit Theorem: Suitable for large samples; involves sample mean and standard deviation.

Example

• Bar Chart Representation: Error bars can represent CI, standard errors, or standard deviations.
• Calculation: Involves sample mean and variance, with results expressed in terms of a Student's t-distribution.

Interpretation

• Long-Run Frequency: CI indicates reliability in repeated sampling.
• Statistical Significance: 95% CI represents values not significantly different from the point estimate at the 0.05 level.

Common Misunderstandings

• Misinterpretation: 95% confidence does not imply 95% probability that the true parameter lies within the interval.
• Example: Misunderstandings about CI in manufactured products like metal rods.

Comparison

• Prediction Intervals: Used for forecasting future observations, different from CI.
• Credible Intervals: Bayesian method, often aligns with CI under non-informative priors.

Examples of Misinterpretation

• Uniform Location: Discusses statistical ideas and misinterpretations relating to CI.
• ( \eta^2 ) in ANOVA: CI behavior as an effect size measure, especially when F-statistic is small.

Specific Distributions

• Binomial, Exponential, Poisson Distributions: Examples of CI applications.
• Regression Analysis: CI for parameters and differences of means.

See Also

• Related topics in statistics such as error bars, Bayesian inference, and prediction intervals.

These notes summarize the key points from a broad overview of confidence intervals, as detailed in the provided Wikipedia article. Understanding these basics aids in applying CI concepts correctly in statistical analysis and avoiding common pitfalls.