In This Lesson Point & Interval Estimation 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 Hypothesis Testing Type I & II Errors Regression Analysis ANOVA & Chi-Square Tests Point & Interval Estimation
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:
State hypotheses: H₀ (null) vs. Hₐ (alternative) Choose α: Significance level (usually 0.05) Compute test statistic: z = (x̄ − μ₀)/(σ/√n) Find p-value: P(observing data this extreme | H₀ is true)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 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
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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.