Units 6-9 are the culmination of AP Statistics — using sample data to draw conclusions about populations through confidence intervals and hypothesis tests.
Estimate ± margin of error. For mean: x̄ ± t*(s/√n). For proportion: p̂ ± z*√(p̂(1-p̂)/n). Interpretation: "We are C% confident that the true [parameter] is between [lower, upper]." NOT "C% probability." Increasing confidence → wider interval. Increasing n → narrower interval. Conditions: random sample, independence (10% condition), normality (n ≥ 30 for means, np̂ ≥ 10 and n(1-p̂) ≥ 10 for proportions).
H₀: null hypothesis (parameter = claimed value). Hₐ: alternative (≠, <, or >). Test statistic: (estimate − null value) / SE. p-value: probability of observing data this extreme if H₀ is true. Decision: if p-value < α (significance level, usually 0.05), reject H₀. Conclusion: "There is/is not sufficient evidence at the α level to conclude [Hₐ in context]." Type I error: reject true H₀ (probability = α). Type II error: fail to reject false H₀ (probability = β). Power = 1 − β.
Two-sample t-test: compare means of two groups. Paired t-test: matched pairs (analyse differences). Two-proportion z-test: compare proportions. Chi-square goodness of fit: observed vs expected distributions. Chi-square test of independence: association between two categorical variables. Conditions: all expected counts ≥ 5. Linear regression t-test: is slope significantly different from 0? H₀: β = 0. Test stat: (b − 0)/SE_b.
The p-value is the probability of obtaining a test statistic as extreme as (or more extreme than) the observed one, assuming the null hypothesis is true. It is NOT the probability that H₀ is true. Small p-value: the observed data would be unlikely under H₀ → evidence against H₀. Large p-value: the observed data is consistent with H₀ → insufficient evidence to reject. Common threshold: α = 0.05 (5%). If p < α, we say the result is "statistically significant." But statistical significance ≠ practical significance — a very large sample can detect tiny, meaningless differences.
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