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Summary of Sampling Distribution in Introductory Statistics
A sampling distribution is the probability distribution of a statistic (e.g., sample mean, variance, or proportion) from a random sample. As estimators are random variables, their distributions help quantify how close a sample statistic is to the true population parameter, crucial for statistical inference.

Key Sampling Distributions
Sample Mean (X̄):
Normal Population, Known Variance: X̄ follows a normal distribution with mean μ and variance σ²/n.
Non-Normal Population (CLT): For large n, the standardized X̄ ((X̄ - μ) / (σ/√n)) approximates a standard normal (N(0,1)).
Normal Population, Unknown Variance: Using sample standard deviation S, ((X̄ - μ) / (S/√n)) follows a t-distribution with (n-1) degrees of freedom.
Sample Variance (S²):
For normal populations, ((n-1)S² / σ²) follows a Chi-squared distribution with (n-1) degrees of freedom, asymmetric and positive-valued.
Sample Proportion (p̂):
For large samples from a Bernoulli/binomial distribution, p̂’s distribution is approximately normal, with mean p and standard deviation √(p(1-p)/n).
Role in Confidence Intervals
Sampling distributions underpin confidence intervals, which estimate a population parameter’s range with a confidence level (e.g., 95%):

Single Mean: Uses normal (known σ²), t-distribution (unknown σ²), or normal approximation (large n).
Single Proportion: Normal approximation for large n.
Two Means: Normal for known/equal variances, t-distribution for unknown/equal variances, or normal approximation for large n.
Two Proportions: Normal approximation for large samples.
Implications
Sampling distributions allow:

Precision Quantification: Standard error determines confidence interval width.
Sample Size Calculation: Ensures desired precision.
Result Interpretation: 95% of repeated intervals contain the true parameter.

Introduction to Statistical Analysis  - Sampling Distribution

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