An Evaluation of Wald Confidence Intervals for the Binomial Proportion: Performance with and without Continuity Correction

Abstract

This study provides a comprehensive evaluation of the performance of the standard Wald and Wald with continuity correction (Wald CC) confidence intervals for the binomial proportion, using simulation techniques across a wide range of population proportions (p) and sample sizes (n). Our evaluation is based on two key criteria: the probability of coverage and the expected interval width for each method. The results demonstrate the inadequacy of the standard Wald confidence interval, which fails to achieve the nominal 95% coverage across virtually all combinations of p and n. It is also highly unstable, even with large samples (e.g., n = 1500, 2000). In contrast, the Wald CC method performs significantly better. It achieves the nominal coverage (and exceeds it) for sample sizes n ≥ 200 when the proportion p is not near the boundaries (i.e., for p between 0.1 and 0.9). Furthermore, its expected interval width is reasonable and decreases appropriately with increasing sample size. However, for extreme proportions (p < 0.1 or p > 0.9), both methods fail catastrophically, and their theoretical bounds often fall outside the [0,1] parameter space. We conclude with a strong recommendation against using the standard Wald interval in practice and advocate for the Wald CC method as a simple and effective alternative, particularly for medium and large samples with non-extreme proportions. For extreme proportions, more effective alternatives recommended.