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.