Confidence and Population Means
The major difficulty in putting the properties of the special batch discussed in Chapter 8 to use is that we had to know a good deal about the population from which the sample was drawn in order to specify the characteristics of the special batch. We knew that the mean of the special batch was the same as the mean of the population and that the standard deviation of the special batch (that is, the standard error of the sample) was the standard deviation of the population divided by the square root of the number in the sample. In real life, however, we do not know either the mean or the standard deviation of the population from which our sample is drawn. Indeed those are precisely the things we are trying to estimate on the basis of a sample. Thus we must find a way to use the special batch without first knowing these characteristics of the entire population. In this chapter we will extend the notion of unusualness of a sample to apply to the more realistic situation in which, instead of having one population and all the possible samples from it, we have one sample and consider the possible populations it might have come from. We will start by asking the question, “How unusual would it be for the sample we actually have to come from a population with a particular mean?” And we will proceed to ask that question about a number of different possible parent populations for our sample.