In This Lesson Point & Interval Estimation 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) Decision: If p-value < α, reject H₀ 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 Example: One-sample z-test 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 Claim: μ = 500. Sample: n = 36, x̄ = 515, σ = 60
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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 1ZNfqBt20kvpTD+UsTvERWArb9DbSFY8IKCenjnv+1M0nBMiMOcV/bKM99GrNg37rDnNA74KoI3o+C66DHgKTu78QrHQw9sqAsBLg7curv13Ix3LKfjHf+rxRp1wacK1M+REGM8fGrtALVVYltJY2lO623m+ULQVjFW9CUG0FsE/8ZreoAsd1fImEqts3q9la890CgKFeUXrV6Q/DcygYyyLf4JdaqDwaPY6Wiw7p48vhYe4un73GtNQmyYTigISiI0Mgm1wTPDtCZtEamdJ1CgZ9CV6yeFxZQJfdwrlgGJG9P4jBM1xf45Xqw0O8s9cW++Onqc2VIradKCg5BpanxlPAUjzspyBIYpUuPVNuaOMAXFEWinvdbmD30walYfIS5a22kDbwGUjivxb18Vd7yKzap6nnreoUpDsDlYBgGKuF9yj4IHezbWNVj5WrwRPeLWPjZJjY+padEwE6dTz9macOd2Or6O9h22/cyo+fVcloBqPRBnjDQcW7HG7ncgdQZTUkcfksuaphENNDMnNym3D/bpitrlm1zDJJYOZGFK4H9ngOIVI21oDTiq0eeHlkh/oGJypZwdbV3Ky9fACU6tcYaoA80TpsI0D/2uMsm8Xx5Xhcv+4dr04nzAlY4UP51YWwzeFI0SX2fQ3Q7RuHpLxwH/tydaT6uABWP4Jd+7DUjeFzNI0JSHlyQjYQcQo6ni3QZ5i9Y4OP2wbmoIqkAZH9RoSXetYhX6cma1dyVu6gqv1HdOCoEHOVnZrQcoNg5HshsHGDOLViosuLCQ9jBQXKGytuUxqf2SfjNhx+zOUPCtfEQH0cDiS3XoQnbJ/F3f51Ek+cZv0gCpEc1gaEArNxEpwQ/VWfirgHXJMvL08nmdjW/k6i5zAIhf6bbn3UcqestWu9h+/VxKPtKSzLNso1D+kh0UrWZRBvsz7dmV3UKAIRK/1xd7mN9IZ/4BmLwdPnVId5iuXFXc6mb9gG9wjBOai8ueoigZ95xuUimMLVwiVWgVyKFd/HNClIoqjJJU8U7DS7//UUQWyujMIvLnDcKusSZiy9bSD5G19je/ 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.
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