Test Assumptions
All parametric statistics have a set of assumptions that must be met in order to properly use the statistics to test hypotheses. The assumptions of the one-sample Z test test are listed below.

• Random sampling from a defined population
• Interval or ratio scale of measurement
• Population is normally distributed

When reading the psychological literature, we can find many studies in which all of these assumptions are violated. Random sampling is required for all statistical inference because it is based on probability. Random samples are difficult to find, however, and psychologists and researchers in other fields will use inferential statistics but discuss the sampling limitations in the article.

We learned in our scale of measurement tutorial that psychologists will apply parametric statistics such as the Z test on approximately interval scales even though the tests require interval or ratio data. This is an accepted practice in psychology.

The central limit theorem tells us that even if the population distribution is unknown, we know that the sampling distribution of the mean will be approximately normally distributed if the sample size is large. This helps to contribute to the Z test being robust for violations of normal distribution.