In This Lesson Point & Interval Estimation Hypothesis Testing 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.
Hypothesis Testing The framework:
State hypotheses: H₀ (null) vs. Hₐ (alternative)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 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 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)cMYRV/ubRDlW8qS57I8rGpjfzLnPkSOvCEQtHSmDROhtUnRmJgknvuFGYB7PvPMQtRpGWETs4dfxGqTHc4sSFClsiLZXecdOgB+4gXHXphz2iAbmFQ3YRgdkqiV+93jetEQc0jZbu4Bm7dR+6wFEsZ73bHt+cFasED5oPpuKTMJCcKX5aHjhXD290+N+yYIPUTWNOD0pmaa3BTJOZUvNxatCZpqS/jHeEGL33jlcXQS6Fm/L8yXN6bG4AanbWdI1rW/WmJcNFllaEYiM2mPcE3euPpwWbNtWNNj9IGSpU0mrUKvifVvxUq8hYNkBXzwAe0DdruqMrdIGAkPwgc/IOiUSacDzmwINmCAZjvkq1DymU2E4TIMmvxuWgt57YRi1g1iqLu/2YTucuPW5ikU9tI9A/6FUpbFET3KvIsV/zwZ/2Miuzl4nyCofuPPJ/HmB3Id6QCJoZ5cRRlDPr1UwuTMMCpayQS/8yRLmRMVkoMwILPHQrxVMJZT0GS2VC93oAGcGco8faeLQVuPHnjvUo5E5z36FAnR8iYLdmQA46ZVodup7TooavHrxPb19I2TMPWd2I33+DnQNyvNZovx2wRpIWTOgUQSBlCEE4WUB32IE9KJ6IeGONXAEqlmPF2ixZHNgUz55o5e2SxfO2mNpEsrg6PzlRzs16DegAX5lWXkGKy8Q8z15W2GuyVFq7Qmqk3rd8ekcktYIe45tZ6np6eBbgOouPhsD29vJT6E/69Rg4uDNo8Gfj+TcIypYnf5abaAuCc8d6YxoB23x0fC7ILWWxRWAgGN0sZN4xZmKtqQfaJYa4UHNwblXGWnNt3Cf+pA915UAFVQZs+aA6dtt6eJntd72py7JhHs8n74rR8b2YbuEPHHI7tiA7KqcF11ClwQFJRKvHYmAL/R90AjG2Mi/NDH4J+hqNHIHpFsJsLClClA5jJC6UWwF3YK2aJTffrKd1N5wwcIeW6RBtwPsnruEBM+uTdUrpz/DSXLzr+Hr2uTCg4ygfPIU933MX+LZCeH92sdhBISoCPRifgqXWw+gnVTkVr1qEpmANe6ANg 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.
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