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Chapter 19: Performing Inferential Statistical Analyses Samples and populations Most research is based on data collected from samples drawn from large populations. Frequently, the sample represents only a small proportion of a population. Nevertheless, as we will show in this chapter, we can use statistics drawn from a sample to estimate the corresponding parameter in a population. We do this by making a statistical inference based on probability theory. This chapter explains the logic behind making statistical inferences and describes a few of the statistical tests used in making inferences. We start with a review of the relations between samples and populations. A sample, you will recall, is a randomly selected part of target population. Using random or chance selection of the sampling elements, and nothing else, allows us as researchers to indicate how well our sample represents the target population. Analysis of data from samples produces statistics. Statistics are used in two ways. First, we use statistics to describe the variables we are studying. Descriptions are usually based on the mean or some other measure of central tendency and the standard deviation for a variable. Statistics are also used to describe relationships among variables. The second use of statistics is to estimate values for variables in the population from which a sample was drawn. The values of variables in a population are referred to as parameters. Because we start with samples, we seldom know the values of parameters for the variables we analyze. By using procedures based on inferential statistics, however, we can use statistics from a sample to estimate the corresponding values of parameters in a population. Whether you make statistical inference or not depends on your research objective. If you are conducting an exploratory study to get ideas for a larger study, you won't be interested in estimating any parameters. Calculation of summary measures limited to describing variables so you better understand them, such as means or percentages, will probably be all you need. Also, many descriptive studies are carried out to identify relationships between variables within the samples without making estimates of corresponding parameters. Most of the research we cited in previous chapters did not involve making statistical inferences. But there may be times when you want to provide an accurate estimate of the parameter for some variable, perhaps the number of children born to some population of married women. In this chapter, we show you how you can use statistics from a sample to do this. A mean and the standard deviation for a variable, for example, can be used to estimate the mean for that variable in the population from which you selected your sample. Estimating parameters requires understanding of probability theory. Probability theory and statistical inference Probability theory is based on the occurrence of random or chance events. It provides theoretical distributions that occur when events occur due to chance and chance only. When we select a probability or random sample, we rely entirely on chance to produce a sample that is representative of the population we want to learn about. Each random sample we select will contain a different combination of respondents or whatever we are sampling. If we drew a very large number of samples, the variation in means or other statistics would form a predictable pattern entirely by chance. We can use this known, predictable pattern, resulting from random variations that occur from one sample to the next, to estimate population parameters. The following illustration shows how we can use sampling variation as a basis for making statistical inferences. |