Confidence intervals are valuable tools in statistical practice for estimating binomial proportions, with the most well-known being the Wald and Clopper-Pearson intervals. However, it is known that these intervals perform poorly in terms of coverage probability and expected mean length, leading to the proposal of alternative intervals in the literature, although these may also have deficiencies. In this work, we investigate the performance of several of these confidence intervals using the parametric family p ^ c = X + c n + 2 c {\widehat{p}}_{c}=\frac{X+c}{n+2c} with c ≥ 0 c\ge 0 to estimate the parameter p p . Rather than using the confidence intervals approach, this analysis is done from the hypothesis tests approach. Our primary goal with this work is to identify values of c c that result in better-performing tests and to establish an optimal procedure.

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