Inference for Categorical Data: Proportions

An estimator is a statistic you use to estimate a parameter; for example, the sample proportion p̂ estimates the population proportion p, and the value you get from one sample is called a point estimate. An estimator is unbiased when, on average, it neither overestimates nor underestimates the parameter, meaning its sampling distribution is centered on the true value.

When the sample is random and the observations are independent, the sampling distribution of p̂ has mean p and standard deviation √(p(1 − p)/n). It is approximately normal when np ≥ 10 and n(1 − p) ≥ 10, and the 10% condition (sample no more than 10% of the population) matters when you sample without replacement.

A one-sample z-interval estimates a population proportion with p̂ ± z* · √(p̂(1 − p̂)/n). The part after ± is the margin of error: the critical value z* multiplied by the standard error (SE). Check the randomization, 10% and normality conditions (at least 10 observed successes and 10 observed failures), and rearrange the margin-of-error formula to find the sample size you need, using p̂ = 0.5 if you have no estimate.

Any one interval either captures the true proportion p or misses it, so you say you're C% confident the interval captures p, and the confidence level means that about C% of intervals built the same way from repeated random samples would capture p. Values inside the interval are plausible, which lets you judge claims, and a higher confidence level makes the interval wider while a larger sample makes it narrower.

A significance test begins with a null hypothesis, H₀: p = p₀ (the "nothing new" claim), and an alternative hypothesis, Hₐ, that p is less than, greater than or not equal to p₀, with the parameter defined in context. The conditions match the interval's, except the normality check uses the null value: np₀ ≥ 10 and n(1 − p₀) ≥ 10.

The p-value is the probability, assuming H₀ is true, of getting a test statistic as extreme as the one you observed, or more extreme, in the direction of Hₐ; with a simulation, it's the share of simulated results that are at least that extreme. The smaller the p-value, the more convincing the evidence for Hₐ, but a large p-value never proves that H₀ is true.

The test statistic is z = (p̂ − p₀) ÷ √(p₀(1 − p₀)/n), and you get its p-value from the standard normal distribution. If the p-value ≤ α (the significance level), reject H₀ and say there is convincing evidence for Hₐ; otherwise, fail to reject H₀, and either way state the conclusion in context without claiming certainty.

A Type I error is rejecting a null hypothesis that is actually true, and a Type II error is failing to reject a null hypothesis that is actually false. The chance of a Type I error is α, the chance of a Type II error is 1 − power, and power (the chance of correctly rejecting a false H₀) goes up with a larger sample, a smaller standard error, a larger α or a true value farther from the null value.

For two independent random samples, or a randomized experiment, the sampling distribution of p̂₁ − p̂₂ has mean p₁ − p₂ and standard deviation √(p₁(1 − p₁)/n₁ + p₂(1 − p₂)/n₂). It is approximately normal when each sample has at least 10 expected successes and 10 expected failures.

A two-sample z-interval estimates p₁ − p₂ with (p̂₁ − p̂₂) ± z* · √(p̂₁(1 − p̂₁)/n₁ + p̂₂(1 − p̂₂)/n₂). The data need to come from two independent random samples or a randomized experiment, and each sample needs at least 10 observed successes and 10 observed failures.

Interpret the interval as a range of plausible values for the difference p₁ − p₂, in context. If every value in the interval is above 0 (or every value is below 0), that's convincing evidence the proportions differ; if 0 is inside the interval, "no difference" is still plausible.

A two-sample z-test checks whether two population proportions differ, starting from H₀: p₁ = p₂ (the same as p₁ − p₂ = 0). For the normality condition you use the combined (pooled) proportion p̂c, the total successes divided by the total of both sample sizes, and check that n₁p̂c, n₁(1 − p̂c), n₂p̂c and n₂(1 − p̂c) are all at least 10.

The test statistic is z = (p̂₁ − p̂₂ − 0) ÷ √(p̂c(1 − p̂c)(1/n₁ + 1/n₂)), and the p-value comes from the standard normal distribution. Compare the p-value with α, then write a conclusion about the difference between the two populations in context.

A chi-square (χ²) statistic measures how far the observed counts are from the counts you'd expect if H₀ were true; χ² distributions take only positive values, are skewed right and get less skewed as the degrees of freedom increase. Use a test for homogeneity to compare one categorical variable's distribution across several populations or treatments, and a test for independence to look for an association between two categorical variables in one population, checking that every expected count is greater than 5.

Each expected count is (row total × column total) ÷ table total, and the test statistic χ² = Σ (observed − expected)² / expected adds up over every cell of the table. Find the p-value from a χ² distribution with df = (number of rows − 1)(number of columns − 1), then compare it with α and state your conclusion in context.