Statistical Inference: A Short Course

7.1 Simple Random Sampling

The fundamental issue addressed by the notion of sampling is to determine when and under what conditions a sample permits a reasonable generalization about a population. While there are many different ways of extracting a sample from a population, we shall primarily engage in simple random sampling. This is because, under random sampling, we can apply the rules of probability theory to calculate the errors associated with using a sample statistic as an estimator for a population parameter. In what follows, we shall always sample “without replacement,” that is, once an element of the population has been selected for inclusion in a given sample, it is no longer eligible to appear again in that sample as the sampling process commences.

How many samples of a given size may be drawn from a given population? Suppose the population is of size N and we are interested in taking samples of size n. Then, since the order of the items within a sample is irrelevant (i.e., an item in the population is either chosen or it is not—we do not care if it was picked first or last), we can theoretically find !/n!(N − n)! possible samples. Hence there are possible ways to obtain a random sample of size n from a population of size N.

How is a random sample defined? A sample is random ...