PROBABILITY THEORY AND INFERENCE I. Introducing inferential statistics. A. Using statistics from probability samples to estimate or test hypotheses about the parameters of the populations from which they are drawn. 1. Statistics and parameters. 2. Estimation and hypothesis testing. B. The importance of probability sampling. 1. In order for probability theory to be applicable, making inference possible, probability samples must be used. 2. In probability sampling every possible sample of a given size has an equal chance of being chosen. C. Types of probability samples. 1. Simple random samples. 2. Systematic random samples. 3. Stratified random samples. 4. Multistage cluster samples. D. Sampling error. 1. Random sampling error. 2. Non-random sampling error. II. Probability. A. Probability is the proportion of times a particular outcome is likely to occur in repeated random sampling on a variable. It can be seen as a proportion of all possible outcomes. B. Probability distributions. 1. A table presenting all possible outcomes and their probability. 2. Graphing probability distributions. III. Sampling distributions. A. A sampling distribution is a theoretical probability distribution of all possible sample outcomes for a statistic given a fixed sample size. B. The sampling distribution of a statistic will be normal with a mean equal to the population parameter of interest and standard deviation equal to the population standard deviation divided by the square root of the sample size if... 1. the population from which the samples are drawn is normally disrtibuted or... 2. the sample size is sufficiently large, say 100 or more. C. This "fact" is called the Central Limit Theorem. It is important because it means that sampling distributions will always be normal (if the sample size is large enough) regardless of the shape of the population distribution. This, in turn, means that the Empirical Rule can be applied to sampling distributions, which, as we will see later, make them important for drawing inferences about populations. D. The standard deviation of a sampling distribution is called the standard error. It can be interpreted as a measure of random sampling error. E. Every sample statistic has a sampling distribution. IV. Distinguishing between and relating sample, population, and sampling distributions. A. The distribution of a sample statistic. B. The distribution of a population parameter. C. The sampling distribution. D. Inference involves using the known empirical sample distribution and the known theoretical sampling distribution to draw conclusions about the unknown empirical population distribution.