Sample size and the width of the confidence interval for mean difference

The width of the confidence interval for mean difference can be viewed as a random variable. Overlooking its stochastic nature may lead to a serious underestimate of the sample size required to obtain an adequate probability of achieving the desired width for the confidence interval. The probability of achieving a certain width can either be an unconditional probability or a conditional probability given that the confidence interval includes the true parameter. We reconciled the difference between the unconditional and conditional probabilities by deriving the lower bound of the conditional probability. Additionally, we used the harmonic mean to determine unequal sample sizes for the confidence intervals for the two-mean comparison and multiple-mean comparisons.