The sample mean should be considered an interval of uncertainty. Although the sample mean appears as a specific number, this appearance is misleading because it is only an uncertain clue to the population mean. To understand the sample mean, this uncertainty must be included.
Statistical theory shows how to make precise this idea of the sample mean as a range of uncertainty. This range of uncertainty can be quantified with two simple formulas; the information in the sample can be used to construct an interval within which the population mean is included with specified confidence. This confidence interval properly represents the sample mean.
As a confidence interval, the sample lifts itself by its bootstraps to the level of the population. Thereby, it provides a foundation for statistical inference.
The logic of the confidence interval is developed along a seven-step road in Section 2.2. Each step on the road involves some fundamental concept. Taken together, the seven steps constitute a chain of reasoning that deserves admiration for internal beauty as well as for practical value. The confidence interval reveals order in disorder; the variability and uncertainty of the sample data are transformed into a precision tool.
The confidence interval can be used as a test of statistical significance—whether experimental manipulations have real effect. Significance tests, however, involve additional concepts, of null hypothesis, false alarm, power, and so forth. Limitations and misuses of significance tests are also discussed.
sample and population significance test sampling distribution null hypothesis variance and standard deviation 2 × 2 decision table confidence interval false alarm and power law of sample size individual differences central limit theorem principle of replication