The two-proportion Z-test (also called the two-sample proportion Z-test) is a statistical hypothesis test for assessing whether two groups differ in the proportion of a binary outcome, in such a significant way that is beyond chance. For example, the proportion of patients responding positively to a treatment in a clinical trial versus control, the defect rate in quality control for two production lines, or the click-through rate in an A/B test of two alternative webpage designs. The test is appropriate when each observation is independent from another, can be classified as a success or failure (i.e., a Bernoulli trial) and the sample sizes are large enough that the sampling distribution of each sample proportion is well approximated by the central limit theorem. Under those conditions the observed difference of sample proportions can be converted to a standardized z-statistic (using a pooled standard error) and compared to the standard normal distribution to obtain p-values or form confidence intervals for the difference in proportions (derived slightly differently). This also allows to determine required sample-size for a minimum-detectable-effect calculations. The test is related to other well known tests such as Pearson's chi-squared test, Fisher's exact test for small samples, and McNemar's test for paired binary data.
Definition The two-proportion Z-test or two-sample proportion Z-test is a statistical method used to determine whether the difference between the proportions of two groups, coming from a binomial distribution is statistically significant. This approach relies on the observation that (for a sufficiently large samples) the sample proportions follow a normal distribution under the Central Limit Theorem, allowing the construction of a z-test for hypothesis testing (a Score test) and confidence interval estimation (a Wald test). It is used in various fields to compare success rates, response rates, or other proportions across different groups.
Hypothesis test The z-test for comparing two proportions is a frequentist statistical hypothesis test used to evaluate whether two independent samples have different population proportions for a binary outcome. Under mild regularity conditions (sufficiently large sample sizes and independent sampling), the sample proportions (which is the average of observations coming from a Bernoulli distribution) are approximately normally distributed under the central limit theorem, which permits using a z-statistic constructed from the difference of sample proportions and an estimated standard error. The test involves two competing hypotheses:
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