Classical inference is utilized to estimate the statistical characteristics of a large population when only a small representative random sample of the population can be obtained. An understanding of classical inference is essential for gaining an appreciation of its strengths and for how Bayesian inference and Dempster-Shafer evidential theory each ameliorate some of its limitations.
Statistical inference uses a number computed from the sample data to make inferences about an unknown number that describes the larger population. In this regard, a parameter is a number describing the population and a statistic is a number that can be computed from the sample data without making use of any unknown parameters. The theory discussed in this chapter is applicable when simple random samples can be gathered. A simple random sample of size n consists of n units from the population chosen in such a way that every set of n units has an equal chance to be the sample actually selected.
More-elaborate sampling designs are often appropriate. For example, stratified random samples are used to restrict the random selection by dividing the population into groups of similar units called strata. Separate simple random samples are then selected from each stratum, as when sampling geographically dispersed populations. Block sample designs are another way to create a group of experimental units that are known before an experiment begins to be similar in some way that is expected to affect the response to the experiment. In a block design, the random assignment of units to treatments or some other influence is performed separately within each block. A third method of restricting random selection is to perform the selection in stages. This is often done when national samples of families, households, or individuals are required. For example, a multi-stage sample design for a population survey may be constructed as follows:
- Stage 1: gather a sample from the 3,000 counties in the United States.
- Stage 2: select a sample of townships within each of the counties chosen.
- Stage 3: select a sample of blocks within each chosen township.
- Stage 4: gather a sample of households within each block.