AP Statistics - Advanced Concepts & Inference

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Power of a Test

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The probability of correctly rejecting a false null hypothesis (1 - Beta). Calculated as P(reject H0 | H0 is false). It increases with larger sample sizes, larger significance levels (alpha), and greater effect sizes. High power is critical for detecting meaningful differences in experimental designs.

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Type I vs. Type II Error Trade-off

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Type I Error (alpha) is rejecting a true H0 (false positive). Type II Error (beta) is failing to reject a false H0 (false negative). Inverse relationship: decreasing alpha (making it harder to reject H0) generally increases beta (risk of missing a real effect), assuming constant sample size.

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Central Limit Theorem (CLT) Conditions

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States that the sampling distribution of the sample mean will be approximately normal if the population is normal OR if the sample size n is large enough (n >= 30). For proportions, the success/failure condition (np >= 10 and n(1-p) >= 10) must be met to assume normality of the sampling distribution.

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Confidence Interval Interpretation

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A 95% confidence interval means that if we took many samples and built an interval from each, approximately 95% of those intervals would capture the true population parameter. It does NOT mean there is a 95% probability that a specific interval contains the parameter.

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Standard Error vs. Standard Deviation

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Standard Deviation (SD) describes the variability of individual data points in a population (sigma) or sample (s). Standard Error (SE) describes the variability of a sampling statistic (like x-bar) and is calculated as SD/sqrt(n). SE decreases as sample size increases.