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Sampling Distributions

5

Introduction

Statistical inference draws conclusions about a population or process from data. It emphasizes substantiating these conclusions via probability calculations because probability allows us to take chance variation into account. We have already examined data and arrived at conclusions many times. How do we move from summarizing a single data set to formal inference involving probability calculations?

The foundation for statistical inference is described in Section 5.1. There, we not only discuss the use of statistics as estimates of population parameters, but also describe the chance variation of a statistic when the data are produced by random sampling or randomized experimentation.

The sampling distribution of a statistic shows how the statistic would vary in identical repeated data collections. That is, the sampling distribution is a probability distribution that answers the question, “What would happen if we did this experiment or sampling many times?” It is these distributions that provide the necessary link between probability and the data in your sample or from your experiment. They are the key to understanding statistical inference.

The last two sections of this chapter study the sampling distributions of two common statistics: the sample mean (for quantitative data) and the sample proportion or count (for categorical data). The general framework for constructing a sampling distribution is the same for all statistics, so we focus on those statistics commonly used in inference. As part of this study, we revisit the Normal distributions and are introduced to two common discrete probability distributions, the binomial and Poisson distributions.