All science rests on evidence from samples. A sample, however, cannot provide an exact picture of the population from which it came. A sample mean, in particular, will always differ from the population mean.
It follows that a sample mean should be represented as an interval of uncertainty. It is only meaningful when accompanied by its likely error from the population mean. a
This view of the sample mean as an interval of likely error is vital in principle. In practice, this view may seem too vague, too indeterminate to be useful. Is it really possible to specify an exact value of likely error?
Remarkably, the answer is yes: Statistical theory has found a way to measure uncertainty. The sow's ear of variability in the sample can be transformed into the silk purse of a confidence interval. The confidence interval epitomizes statistical inference—and provides a tool for empirical analysis.
The prototypical problem of statistics is to use sample data to make inferences about populations. This requires an idealization in which we consider a random sample of elements drawn from some specified population. The elements are assumed to be independent: Knowledge of any one element tells nothing about any other element; each added element carries equal information.____________________