T test as a parametric statistic

Abstract

In statistic tests, the probability distribution of the statistics is important. When samples are drawn from population N (µ, σ ²) with a sample size of n, the distribution of the sample mean X̄ should be a normal distribution N (µ, σ ²/ n). Under the null hypothesis µ = µ ₀, the distribution of statistics $\mathrm{z}=\frac{\overline{\mathrm{X}}-{\mathrm{\mu }}_0}{\mathrm{\sigma }/\sqrt{\mathrm{n}}}$ should be standardized as a normal distribution. When the variance of the population is not known, replacement with the sample variance s ² is possible. In this case, the statistics $\frac{\overline{\mathrm{X}}-{\mathrm{\mu }}_0}{\mathrm{s}/\sqrt{\mathrm{n}}}$ follows a t distribution ( n-1 degrees of freedom). An independent-group t test can be carried out for a comparison of means between two independent groups, with a paired t test for paired data. As the t test is a parametric test, samples should meet certain preconditions, such as normality, equal variances and independence.