" .... t-test is appropriate when the size of the sample is small, in the sense that n < 30 (refer to central limit theorem)..
The t-distribution is typically used to study the mean of a population, rather than to study the individuals within a population. In particular, it is used in many cases when you use data to estimate the population mean — for example, using the sample mean of n=20.
The connection between the normal distribution (z-distribution) and the t-distribution is that the t-distribution is often used for analyzing the mean of a population if the population has a normal distribution (or fairly close). Its role is especially important if your data set is small.
When statisticians use the term t-distribution, they aren’t talking about just one individual distribution.
There is an entire family of specific t-distributions, depending on what sample size is being used to study the population mean. t-distribution is distinguished by its degrees of freedom. In situations where you have one population and your sample size is n, the degrees of freedom for the corresponding t-distribution is n – 1. For example, a sample of size 10 uses a t-distribution with 10 – 1, or 9, Situations involving two populations use different degrees of freedom.."
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