Introduction

Statistical Science, through Inferential Statistics, provides us with the necessary tools to draw conclusions about the parameters of a population based on data derived from samples. One of the most fundamental tools of this process is the Confidence Interval. Understanding this concept is particularly important, as it allows for the correct evaluation of estimates derived from data, while also protecting against frequent misinterpretations that can lead to incorrect scientific or practical conclusions.

Definition of Confidence Interval

A confidence interval is defined as a range of values obtained from sample observations that is estimated to contain the true value of an unknown population parameter with a certain probability. For example, a 95% confidence interval for the population mean implies that if the same experiment were repeated many times and a confidence interval was calculated each time, then approximately 95% of these intervals would contain the true mean. Most commonly, confidence levels of 95% or 99% are chosen, as they provide a balance between reliability and the practical usefulness of the estimates.

Confidence Level and Significance Level

The Confidence Level is the degree of certainty with which we can claim that the constructed interval contains the true parameter value. In contrast, the Significance Level is defined as the complement of the confidence level, meaning it equals one minus the confidence level. Thus, for a 95% confidence interval, the significance level is 5%, which means there is a 0.05 probability that the interval does not contain the true value. The smaller the significance level, the greater the certainty of the estimate. This relationship directly influences both the width of the interval and the interpretation of the results.

Construction of Confidence Intervals

The construction of a confidence interval is based on probability theory and on the properties of the distributions that describe the data. The interval takes the form of a lower and an upper bound, that is, [A, B]. The width of the interval depends on three main factors. The first is the sample size, since the larger the sample, the narrower the interval and therefore the greater the precision of the estimate. The second is the variability of the data, because high dispersion leads to wider intervals that express greater uncertainty. The third is the selected confidence level, as higher confidence levels require larger intervals to ensure greater probability of containing the true value. In contrast with a point estimate, which provides only a single value without expressing the uncertainty that accompanies it, the confidence interval reflects this uncertainty through its width and provides a more complete picture of the estimate.

Misinterpretations of Confidence Intervals

Despite their significant usefulness, confidence intervals are often misinterpreted. A common misunderstanding is that 95% of the data lie within the interval, which is not true, since the interval concerns the estimate of the parameter and not the distribution of the individual sample observations. Another incorrect interpretation is that the interval represents the range of plausible values for the sample, while in reality it is an estimate of the possible values of the population parameter. Finally, it is not correct to claim that there is a 95% probability that the parameter lies within the specific interval calculated from one sample. The parameter is fixed and unknown, while the interval is what changes from sample to sample. The correct interpretation is that, across many repetitions of the same procedure, 95% of the intervals constructed will contain the true value.

Conclusions

Confidence intervals are a fundamental tool of statistical analysis and surpass point estimates because they capture the degree of uncertainty that accompanies estimations. Through their width, they provide crucial information about the accuracy and reliability of the results. However, their correct interpretation is essential in order to avoid misconceptions that could lead to false conclusions. Their proper use is a cornerstone of scientific research and contributes to well-founded decision-making in all fields where statistical methods are applied.