8.4:
Choosing Between z and t Distribution
The z and t distributions can estimate the mean of a population using a sample statistic. But how does one choose the appropriate distribution for a given dataset?
The z distribution is preferred for populations with a known standard deviation that is normally distributed or populations that have a sample size greater than 30.
However, the Student t distribution is preferred if the population standard deviation is unknown for a normally distributed population or if the population has a sample size greater than 30.
Symmetrically distributed datasets with very large sample sizes show less variability. For such datasets, the population mean estimated by both the z and t distributions are similar.
The z and t distributions are limited to random samples drawn from normally distributed populations. Thus, they cannot estimate the population mean for samples drawn from voluntary sample responses, convenience sampling, or skewed or unknown population distributions.
Therefore, nonparametric statistics or computer bootstrapping methods are used for populations that are not normally distributed and those populations that have a sample size less than or equal to 30.
8.4:
Choosing Between z and t Distribution
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.