Difference of independent sample proportions ($\hat{\delta }={\hat{p}}_1-{\hat{p}}_0$)

For two independent samples of Bernoulli random variables, the difference of their sample proportions estimates the difference of population probabilities. Since the samples are independent,Substitution of the sample proportions gives a standard error for a large-sample Wald confidence interval. This interval is a descriptive comparison of the specified groups; a causal effect needs further assumptions. Sparse counts and probabilities near zero or one can make the normal approximation poor. Testing equality can instead use a null-pooled estimate of the common probability for the test's standard error.