The true probability of a confidence interval

The true probability of a confidence interval

Summary

The calculation of a confidence interval, which together with the hypothesis testing is the best known procedure of inferential statistics, has as result the probability that a certain statistical parameter is contained in a certain part of the real line. However, this result does not enjoy of unanimity because it is widely believed the not be strictly a probability and that must be called only confidence. To this is added the perplexity of being able to replace, as is highlighted in the article, the said probability with many other equally reliable.
These uncertainties are tackled by distinguishing, among all those of the same event, only one probability true and therefore not merely conventional, and then choosing, as result of the determination of a confidence interval, the true inherent probability which, although it is not exactly calculable, however is unlimitedly approximable.
For this purpose, it is preliminarily dedicated much care in defining the symbology and the concepts of logic and set theory needed for the subsequent deductions, substantially taking again notions of [1] such as the original algorithmic definitions of relations and operations between sets, the unusual formulation concerning the equality between the intersection of products and the product of intersections, and an expanded form of the important tautology that includes the known “law of contraposition”.
The treatment of events and probability exposed in [1] is summarized, simplified and integrated by new decisive positions. It is thoroughly analyzed the event constituted by the happen an unknown constant into a certain part of the real line and its probability, because fundamental for the treatment of the confidence interval which is then deduced and specified in detail for the two cases, of great importance in the experimental sciences, that are had when the statistical parameter is the mean or the variance of a normal random variable.

Keywords: confidence interval, probability, statistics, propositional logic, set theory, combinatorics.

INDEX

1 INTRODUCTION

2 PRELIMINARIES OF LOGIC AND SET THEORY

3 EVENTS AND PROBABILITY

3.1 Composite events

3.2 Probability

3.3 An application of composite events

4 THE PROBABILITY OF AN UNKNOWN CONSTANT

4.1 A confirmation

5 THE CALCULATION OF THE CONFIDENCE INTERVAL

5.1 The mean of a normal random variable