Large and Small Sample Problems for Normal or t-Test

Sample size plays a fundamental role in statistical inference using normal and t-tests, yet inappropriate sample sizes can lead to serious inferential distortions. Large samples tend to produce artificially small p-values and overly narrow confidence intervals, resulting in test bias and potentially misleading conclusions. In contrast, small samples often yield unstable variance estimates, reducing the reliability and power of statistical tests. This paper addresses both extremes within a unified framework. For large samples, we develop the Random Group Method (RGM) and a correction factor approach to mitigate bias by stabilizing variability across subgroups. For small samples, we introduce pseudo-sample expansion techniques, emphasizing a mean-based method that preserves expectation while reducing variance inflation caused by duplication. A key contribution of this work is the formulation of a relative variance (RV) criterion for determining a “good” sample size. We show that the optimal sample size is not a single value but lies within an interval over which statistical inference remains stable. Theoretical results are supported by numerical examples illustrating the relationship between sample size, variance control, and hypothesis testing behavior. The proposed methods provide practical tools for improving statistical inference in both traditional settings and modern large-scale data applications, including artificial intelligence and survey analysis.