Statistical Inference

Point & Interval Estimation

A 95% CI means: if we repeated the sampling many times, about 95% of the intervals would contain the true parameter. This frequentist interpretation connects to probability theory. The margin of error shrinks as n grows — reflecting the limit behavior of estimation.

Hypothesis Testing

The framework:

1. State hypotheses: H₀ (null) vs. Hₐ (alternative)
2. Choose α: Significance level (usually 0.05)
3. Compute test statistic: z = (x̄ − μ₀)/(σ/√n)
4. Find p-value: P(observing data this extreme | H₀ is true)
5. Decision: If p-value < α, reject H₀

Type I & II Errors

• Type I (α): Rejecting H₀ when it's true (false positive)
• Type II (β): Failing to reject H₀ when it's false (false negative)
• Power = 1 − β: Probability of correctly rejecting a false H₀

Increasing sample size increases power without inflating α. These trade-offs are fundamental to experimental design.

Regression Analysis

Regression finds the line of best fit using calculus optimization (minimizing the sum of squared residuals). For multiple predictors, matrix algebra gives the solution: b = (XᵀX)⁻¹Xᵀy.

ANOVA & Chi-Square Tests

ANOVA (Analysis of Variance) tests whether means differ across groups — it generalizes the t-test. The F-statistic = MS_between / MS_within. Chi-Square tests independence in contingency tables and goodness-of-fit for categorical data.