Comparing the Overlapping of Two Independent Confidence Intervals with a Single Confidence Interval for Two Normal Population Parameters

Abstract

Two overlapping confidence intervals have been used in many sources in the past 30 years to conduct statistical inferences about two normal population means. Several authors have examined the shortcomings of Overlap procedure in the past 13 years and have determined that such a method completely distorts the significance level of testing the null hypothesis of two normal population means and reduces the statistical power of the test. Nearly all results for small sample sizes in Overlap literature have been obtained either by simulation or by somewhat inaccurate formulas, and only large-sample (or known-variance) exact information has been provided. Nevertheless, there are many aspects of Overlap that have not yet been presented in the literature and compared against the standard statistical procedure. This paper will present exact formulas for the % overlap, ranging in the interval (0, 61.3626%] for a 0.05-level test, that two independent confidence intervals (CIs) can have, but the null hypothesis of equality of two population means must still be rejected at a pre-assigned level of significance for sample sizes ? 2. The exact impact of Overlap on the?alpha-level and the power of pooled-t test will also be presented. Further, the impact of Overlap on the power of the F-statistic in testing the null hypothesis of equality of two normal process variances will be assessed. Finally, we will use the noncentral t distribution, which has never been applied in Overlap literature, to assess the Overlap impact on type II error probability when testing the equality of two population means for sample sizes equal two.