In This Lesson Point & Interval Estimation Hypothesis Testing 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 Type I & II Errors 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 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.
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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) Decision: If p-value < α, reject H₀ 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 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.