AP Statistics Unit 3 Cheatsheet: Inference for Categorical Data: Proportions

• Parameter describes a population; statistic describes a sample and estimates its matching parameter. p̂ = x/n estimates p.
• Unbiased estimator: its sampling distribution is centered at the parameter; μ_p̂ = p.
• For independent observations: σ_p̂ = √[p(1−p)/n]. Larger n → less variability.
• Approx. Normal when np ≥ 10 and n(1−p) ≥ 10.
• Random sample + 10% condition (n ≤ .10N) justify independence when sampling without replacement.

Confidence interval for p

• Conditions: random; 10%; large counts using p̂: np̂ ≥ 10 and n(1−p̂) ≥ 10.
• Interpret: “We are C% confident that the interval (a, b) captures the true proportion of [population] who [response].”
• Repeated-sampling meaning: about C% of intervals made this way capture p—not C% of sample values or a probability about a fixed p.
• MOE = z*·SE. Higher confidence → wider; larger n → narrower, proportional to 1/√n.
• Plan n: n = (z*)²p*(1−p*)/MOE²; if no prior estimate, use p* = .5; round up.

Hypothesis test for p

• Conditions: random; 10%; large counts using p₀: np₀ ≥ 10 and n(1−p₀) ≥ 10.
• H_a uses <, >, or ≠ from the research question. Parameters and hypotheses must be in context.
• p-value: assuming H₀ is true, probability of a statistic as extreme or more extreme in H_a’s direction.
• If p-value ≤ α, reject H₀; otherwise fail to reject. Conclude with “convincing evidence” or “not convincing evidence” for H_a.

• Power = P(reject H₀ when a specified alternative is true) = 1−β.
• Power rises when n ↑, α ↑, variability/SE ↓, or true p is farther from p₀.
• Choose α by weighing real-world Type I consequences; choose n partly by Type II consequences.