Effect of Positively Skewed Distribution on the Two sample t-test: Based on Chi-square Distribution

This research examines the effect of positively skewed population distribution on the two sample t-test through simulation. For simulation work, two independent samples were selected from the same chi-square distributions with 3, 5, 10, 15, 20, 30 degrees of freedom and sample sizes 3, 5, 10, 15, 20, 30, respectively. Chi-square distribution is largely skewed to the right at small degrees of freedom and getting symmetric as the degrees of freedom increase. Simulation results show that the sampled populations are distributed positively skewed like chi-square distribution with small degrees of freedom, the F-test for the equality of variances shows poor performances even at the relatively large degrees of freedom and sample sizes like 30 for both, and so it is recommended to avoid using F-test. When two population variances are equal, the skewness of population distribution does not affect on the t-test in terms of the confidence level. However even though for the highly positively skewed distribution and small sample sizes like three or five the t-test achieved the nominal confidence level, the error limits are very large at small sample size. Therefore, if the sampled population is expected to be highly skewed to the right, it will be recommended to use relatively large sample size, at least 20.