Random Sampling

Simple Random Sampling

Simple random sampling is the simplest random sampling scheme. There are two types of simple random sampling: with or without replacement. For simple random sampling without replacement, (n distinct units are selected from a population of N units so that every possible sample has the same chance of being selected). Thus the probability of selecting any individual sample s of n units is

Sampling without replacement is often preferred because it provides an estimate with smaller variance. However, even though for a finite population, when a sample has been selected, it does not [Page 1211]provide additional information to be included in the sample again. One practical advantage to sample with replacement is there that is no need to determine whether a unit has already been sampled. In addition, when sampling with replacement, each drawing of each sample is independent, and the probability formulas for the samples are usually easier to derive. For simple random sample with replacement, a unit may be selected more than once, and each draw of a unit is independent: Each unit in the population has the same probability of being included in the sample. When the sample size is much smaller than the population size, the two sampling schemes are about the same. Otherwise, it is important to distinguish whether the simple random sampling is with or without replacement since the variance for statistics using these two schemes is different, and thus most inference procedures for these two schemes will be different. For example, in estimating the population mean under simple random sampling without replacement, the estimated variance for the sample mean is

where s2 is the sample variance.

Note that the factor

is called the finite population correction factor. Under simple random sampling with replacement, there is no need for this factor, and the estimated variance for the sample mean is simply

In designing a survey, one important question is how many to sample. The answer depends on the inference question one wants to answer. If one wants to obtain a confidence interval for the parameter of interest, then one can specify the width and level of significance of the confidence interval. Thompson (2002), chapter 4, provides an excellent discussion for determining sample sizes when one is using a simple random sample.

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