The z and the Student t distribution estimate the population mean using the sample mean and standard deviation. However, to decide which distribution to use for a calculation, one needs to determine the sample size, the nature of the distribution, and whether the population standard deviation is known. If the population standard deviation is known and the population is normally distributed, or if the sample size is greater than 30, the z distribution is preferred. The Student t distribution is preferred when the population standard deviation is unknown, and the population is normally distributed; or if the sample size exceeds 30.
It is important to note that for a sample with a size less than 30, drawn from a skewed or unknown distribution, neither the z nor t distribution can be used. Therefore, z and t distributions cannot accurately estimate the population mean for samples drawn from voluntary responses, convenience sampling, or skewed or unknown population distributions. One must employ nonparametric statistical methods such as bootstrapping for categorical data or when the sample size is small, i.e., less than 30.
