Statistical Inference

Make evidence-based conclusions about populations using sample data.

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

Point & Interval Estimation

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
Confidence interval for the mean:
x̄ ± z*(σ/√n)  (known σ)
x̄ ± t*(s/√n)  (unknown σ, use t-distribution)

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:

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

Example: One-sample z-test

Claim: μ = 500. Sample: n = 36, x̄ = 515, σ = 60

z = (515 − 500)/(60/√36) = 15/10 = 1.5

p-value ≈ 0.134 > 0.05 → fail to 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

Simple linear regression: ŷ = b₀ + b₁x
b₁ = Σ(xᵢ − x̄)(yᵢ − ȳ)/Σ(xᵢ − x̄)²
b₀ = ȳ − b₁x̄
R² = 1 − SS_res/SS_tot

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.

Modern statistics increasingly uses computational methods: bootstrapping, permutation tests, and Bayesian approaches. These still rely on the probability and descriptive foundations covered earlier, but add computational power to handle complex real-world data.