Random Sampling

Three concepts that are relevant to understanding “random sampling” are population, sample, and sampling. A population is a set of elements (e.g., scores of a group of prisoners on a personality scale, weights of newborns of mothers younger than 18, longevity of smokers in New York City, and changes in the systolic blood pressure of hypertensive patients exposed to relaxation training) defined by the researcher's interests. Populations can be defined narrowly or broadly and can consist of a few or many elements. For example, a researcher concerned with effects of relaxation training on systolic blood pressure might be interested only in pre- to post-treatment systolic blood pressure changes (i.e., pre–post) in the small group of patients included in her study, or she might be interested in the efficacy of relaxation training in the collection of all persons who could be classified as “elderly obese hypertensive patients” by some operational definitions of these terms.

A sample is a subset of a population, and sampling refers to the process of drawing samples from a population. Sampling is generally motivated by the unavailability of the entire population of elements (or “data”) to the researcher and by her interest in drawing inferences about one or more characteristics of this population (e.g., the mean, μ, or variance, σ2, of the population). Thus, in the hypertension example, the researcher might be interested in estimating from her sample data the mean (μ) or variance (σ2) of pre- to post-treatment changes in systolic blood pressure of the population consisting of persons who fit the operational definition of “elderly obese hypertensive patients.” She might also be interested in using the sample data to test (the tenability of) some hypothesis about μ, such as the hypothesis that the average change in systolic blood pressure (μ) in the population of hypertensives exposed to relaxation training is zero. This hypothesis will be called the “null hypothesis” and will be abbreviated “H0” below. Because characteristics of a population, such as μ and σ2, are defined as “para-meters” of the population, the principal objectives of random sampling can often be described in terms of (a) estimation of population parameters and (b) testing of hypotheses about population parameters. Parameters that are often of interest are means (μs) and linear combinations of means that provide information about mean differences, trends, interaction effects, and so on.

Simple random sampling refers to a method of drawing a sample of some fixed size n from the population of interest, which ensures that all possible samples of this size (n) are equiprobable (i.e., equally likely to be drawn). Sampling can be carried out with [Page 817]replacement or without replacement. Sampling is with replacement if, in the process of drawing the n elements of the sample consecutively from the population, each element is replaced in the population prior to drawing the next element. For example, if the population of interest consisted of Minnesota Multiphasic Personality Inventory Depression scores of 60, 65, and 70, drawing a sample of size n = 2 with replacement involves replacement of the first-drawn score prior to making the second draw. Sampling is without replacement in this example if the first-drawn score is not replaced prior to making the second draw. Therefore, there would be nine possible samples of size n = 2 (in this example) when sampling was with replacement, as shown

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