T test as a parametric statistic

In statistic tests, the probability distribution of the statistics is important. When samples are drawn from population N (mu, sigma(2)) with a sample size of n, the distribution of the sample mean X-should be a normal distribution N (mu, sigma(2)/n). Under the null hypothesis mu = mu(0), the distribution of statistics z = X-mu 0/sigma/root 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 X-mu 0/s root 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.