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Abstract:
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Consider the established method for constructing confidence intervals (CIs) for the mean of normal data with unknown variance. First, we use the unbiased variance estimator with (n - 1) in the denominator, instead of the biased maximum likelihood (ML) estimator with n in the denominator (n = sample size). Second, we use the t-distribution instead of the standard normal distribution. Third, we implicitly benefit from the continuity of normal data. By contrast, we forget these three key lessons when we construct binomial CIs. First, we use the biased ML variance estimator, p(1 - p)/n, instead of the unbiased estimator, p(1 - p)/(n - 1), (p = sample proportion). Second, we do not use the t-distribution, despite the fact that the true variance is unknown. Third, we seldom adjust for discreteness. By applying these three lessons, however, we can obtain binomial CIs with near nominal coverage.
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