A Comparison Of Usual t-Test Statistic and Modified t-Test Statistics on Skewed Distribution Functions

When the sample size n is small, the random variable T= √n(\overline{X} – μ)/S is said to follow a central t distribution with degrees of freedom (n – 1), where \overline{X} is the sample mean and S is the sample standard deviation, provided that the data X ~ N (μ, σ2). The random variable T can be used as a test statistic to hypothesize the population mean μ. Some argue that the t-test statistic is robust against the normality of the distribution and claim that the normality assumption is not necessary. In this article we will use simulation to study whether the t-test is really robust if the population distribution is not normally distributed. In particular, we will study how the skewness of a probability distribution will affect the confidence interval as well as the t-test statistic