Inference for Categorical Data: Proportions
The logic of inference
Inference starts with a null hypothesis H₀ (typically p = p₀) and an alternative H₁ (one- or two-sided). A p-value is the probability, assuming H₀ is true, of observing a statistic at least as extreme as the one computed. It is not the probability that H₀ is true - a frequent trap. Compare the p-value to α: reject H₀ when p-value < α. The conclusion is always about the parameter in context, never about the sample.
One-proportion z-procedures
Conditions: random sample, independence (10% condition when sampling without replacement), and large counts (np₀ and n(1−p₀) ≥ 10 for the test; n·p̂ and n(1−p̂) ≥ 10 for the interval). Test statistic: z = (p̂ − p₀)/√(p₀(1−p₀)/n). Confidence interval: p̂ ± z*√(p̂(1−p̂)/n). Interpret a 95% interval as: "We are 95% confident the true proportion lies between __ and __" - not "95% probability the parameter is in this interval."
Two-proportion procedures and errors
For comparing groups, use p̂₁ − p̂₂ with standard error √(p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂); pooled proportions apply only to the test, not the interval. A Type I error rejects a true H₀ (probability α); a Type II error fails to reject a false H₀ (probability β). Power = 1 − β, and increases with larger n, larger effect size, or larger α.
Worked example. In 200 trials, p̂ = 0.60. Testing H₀: p = 0.50 vs. H₁: p > 0.50 gives z = 0.10/√(0.25/200) ≈ 2.83, p-value ≈ 0.002, so reject H₀ at α = 0.05.