In simple random sampling, the probability that each unit will be sampled is the same. Sometimes, estimates can be improved by varying the probabilities with which units are sampled.
For example, we want to estimate the number of job openings in a city by sampling firms in that city. Many of the firms in the city are small firms. If one uses s.r.s, the size of a firm is not taken into consideration and a typical sample will consist of mostly small firms. However, the number of job openings is heavily influenced by large firms.
Thus, we should be able to improve the estimate of the number of job openings by giving the large firms a greater chance to appear in the sample, for example, with probability proportional to size or proportional to some other relevant aspects.
- Selection probabilities
On each draw, the probability that a given population unit will be selected is denoted as \(p_i\), i = 1, 2, 3, ….., N.
Suppose that sampling is with replacement, the probability of selecting the ith unit in the population is \(p_i\).
If the selection probabilities are unequal, the sample mean is not unbiased for the population mean and the sample total is not unbiased for the population total.
Example: If larger firms are sampled with higher probability, the sample mean for job openings will be biased upward.