Confidence Interval

Interval Estimation Versus Point Estimation

The purpose of inferential statistics is to infer properties about an unknown population parameter using data collected from samples. This is usually done by point estimation, one of the most common forms of statistical inference. Using sample data, point estimation involves the calculation of a single value, which serves as a best guess or best estimate of the unknown population parameter that is of interest.

Instead of a single value, an interval estimate specifies a range within which the parameter is likely to lie. It provides a measure of accuracy of that single value. In frequentist statistics, confidence intervals are the most widely used method for providing information on location and precision of the population parameter, and they can be directly used to infer significance levels. Confidence intervals can have a one-sided or two-sided confidence bound. They are numerical intervals constructed around the estimate of the unknown population parameter. Such an interval does not directly infer a property of the parameter; instead, it indicates a property of the procedure, as is typical for a frequentist statistical procedure.

The American Psychological Association’s Publication Manual strongly recommends the use of confidence intervals for reporting statistical analysis results. In fact, in the literature, it has been concluded that confidence intervals and null hypothesis significance testing are two approaches to answer the same research question. They give accessible and comprehensive point and interval information to support substantive understanding and interpretation. As George Casella and Roger L. Berger pointed out, in general, every confidence interval corresponds to a hypothesis testing and vice versa. Whenever possible, researchers should base discussion and interpretation of results on both point and interval estimates whenever possible.

Brief History of Confidence Intervals

In the early 19th century, Pierre-Simon Laplace and Carl Friedrich Gauss had already recognized the need for interval estimation to provide information about measures of accuracy. However, the term confidence intervals was not used until Jerzy Neyman’s presentation before the Royal Statistical Society in 1934. In the appendix of this paper entitled “On the Two Different Aspects of the Representative Method,” Neyman proposed a straightforward way to create an interval estimate and to determine how accurate the estimate is based on sample data. He called this new procedure confidence intervals and the ends of the confidence intervals confidence bounds. Also, the arbitrarily defined values termed confidence coefficients indicated how frequently the observed interval obtained from sample data contains the true population parameter if the experiment is repeated. Nowadays, a confidence coefficient is often referred to as a confidence level for its relation to null hypothesis significance testing. Neyman finally addressed the theory of confidence intervals extensively in 1937 in “Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability.” In this paper, the mathematical assumptions, derivations, and proofs provided the philosophical and statistical foundation for confidence intervals.

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