# Confidence Interval

Confidence interval is one of the common concepts used in statistical studies. There are quite a number of cases where this concept yields great results. This is used to obtain a value which is almost equal to the universal value of any given mathematical problem.

Sometimes, different samples provide different estimates at different times. These generally differ from the universal value because of the sampling error, i.e., the sample chosen is not a true replica of the universal data-set. Any random sample selected and estimated by researchers might give out a different result from that of another random sample. So, confusion arises on which one of the values is right and which gives us the best possible estimate. A question arises on the precision of the sample estimates as well.

In such cases, what is usually done is a general range of values is determined within which the true universal value is most likely to fall. At the same time, a percentage of surety is also determined on the universal value being there in the range of estimated values.

The range of values is called confidence interval. It is basically an interval estimate of a sample or more precisely, population parameter and the surety or probability of the universal value falling within the range is called the confidence level. It is used for the reliability purpose of an estimate. More the percentage, wider is the confidence interval. In fact, it is quite possible that the confidence interval estimate in some cases can range from minus to plus infinity.

Its role in the frequent statistics can be compared with the role of credibility intervals in Bayesian statistics but worthy to be noted here that both forms are absolutely different mathematically and have two extreme variant interpretations.

A simple example would be best to clarify the confidence interval concept. It can be used to determine the reliability of survey results. In an election voting-intentions poll, if a survey says 40% of respondents intend to vote for a certain party, the confidence interval estimate would say that it is 95% sure that 36% to 44% of the respondents would vote for the certain party. In such cases, low CIs are more reliable than higher CIs. All other things being equal in the survey, one of the factors that play a very important role is the size of the sample estimated. Confidence intervals and intervals estimates are a common term in case of quantitative studies.

A statistic presented with confidence interval, that is supposed to be statistically significant, would provide value that comes from a test, performed at significant level of 100% minus the confidence level of the interval. These are more informative than the simple hypothetic tests since they state a range of certain credible values for the unknown factor or parameter.

Properties of a Quality CI: At the time of application of fairly standard procedures in statistics, there will be certain standard ways to construct the confidence interval. To meet the industry standards of the statistical procedure, a confidence interval should meet certain requirements in regards to the validity, optimality and invariance.

**• Validity:** Validity makes sure that the confidence level or the probability measurement of the confidence interval should hold to be exact or to be at a good level of approximation. In other words, it has to be perfect or nearly perfect.

**• Optimality:** Optimality ensures that the procedure makes use of as much of the information that can be obtained from the data-set to provide a high confidence level. This ensures the possibility of a confidence interval with shorter width and a high confidence level. Thus, it is also a measurement of the standard of the confidence interval. But it should be noted, that even with half the data set, a quality and fairly exact confidence interval is possible.

**• Invariance:** In certain applications, the estimate is more like an estimate of the logarithm of the sample. So, in order to provide a more accurate estimated range, it is desirable that the confidence interval of the sample survey data gives equivalent estimates as the confidence interval of the logarithm of the sample survey data. More specifically, the values where the latter confidence interval ends should be equivalent to the logarithms of the values of the former confidence interval ends.