CDFt

CDFt stands for the Cumulative Distribution Function of the t-distribution. The t-distribution, also known as Student's t-distribution, is a probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown. CDFt is a function that, for a given value x and a specified number of degrees of freedom (often denoted by ν or df), calculates the probability that a random variable following the t-distribution with that many degrees of freedom will take on a value less than or equal to x.

Mathematically, CDFt(x; ν) = P(T ≤ x), where T is a random variable following the t-distribution with ν degrees of freedom. The degrees of freedom determine the shape of the t-distribution; as the degrees of freedom increase, the t-distribution approaches the standard normal distribution. The CDFt is a cumulative probability function, meaning it provides the area under the probability density function of the t-distribution from negative infinity up to the specified value x.

CDFt is a fundamental tool in statistical inference, particularly in hypothesis testing and confidence interval construction for means. For example, it is used to calculate p-values in t-tests. When performing a t-test, the calculated t-statistic is compared to the CDFt to determine the probability of observing such a statistic (or a more extreme one) if the null hypothesis were true. This probability is the p-value. Similarly, it is used to find the critical values for confidence intervals.