Why is a t-statistic, as opposed to a z-statistic, used to test small populations?

Answer 1

The t-test is used to find a t statistic when you don't have the population standard deviation to plug into the equation (you just have the standard deviation of your sample). The z-test is used when you do have the population standard deviation. If there is a normal distribution the mean of the population is equal to the mean of your sample.

Answer 2

to add to the answer above, a t-statistic also requires knowing the degrees of freedom, which simply your sample size, n, minus 1. Looking up the Table of Values for t-statistics require you needing the degrees of freeodm, as well as the right-tail area of the distribution.

For example, if the confidence level is 90%, then 10% is the level of significance. And of this 10%, 5% belongs in the left tail, and the other 5% belongs in the right tail of the distribution.

so for one sample t-tests: t = x - u / {s/Sqrt(n)}, where x = sample mean, u = population mean, s = sample standard deviation, and n = sample size

hope that helped.