Confidence Interval

Most statistical analysis is conducted to reach some conclusion or decision about one or more parameters associated with a population of interest (statistical inference). There are two types of estimators used to assist in reaching a conclusion: point estimators and interval estimators. If the sampling distribution of the point estimator is known, then a likely range of values can be computed for the parameter being estimated. This range is called an interval estimator or confidence interval.

Before proceeding to more detail, here are some terms specific to confidence intervals. Confidence limits or confidence bounds are the upper and lower values of the range for the parameter given by the confidence interval. These limits are obtained from a sample and are random variables. The pivotal quantity is the point estimate used to estimate the population parameter and is the center of the confidence interval. The half-width of the confidence interval is the distance from the pivotal quantity to one of the limits and is a positive value. The desired significance level for the confidence interval is denoted by a (alpha), and is given by a number between 0 and 1. Typical values for a are between 0.01 and 0.05. The confidence coefficient is calculated as (1 − a), and is sometimes called the coverage probability or inclusion probability (although these terms are misleading, as explained later). Typical values for the confidence coefficient are between 0.95 and 0.99. The confidence level or degree of confidence is the confidence coefficient expressed as a percentage, 100(1 − a)%. Typical values for the confidence level are between 95% and 99%. The standard error or standard deviation of the sampling distribution is used in calculating the confidence interval and depends on the particular point estimator used as well as the sampling method used.

Since point estimates do not provide a means for assessing the reliability or confidence placed on them, confidence intervals are preferred. They require no [Page 226]additional information about reliability since it is provided by the degree of confidence and the half-width of the interval. The goodness of an interval estimation procedure can be determined by examining the fraction of times in repeated sampling that interval estimates would contain the parameter to be estimated. This fraction is called the confidence coefficient. The confidence coefficient is specified by the researcher, and it expresses how much assurance he or she wishes to have concerning whether the interval estimate encompasses or ‘covers’ the parameter of interest. If 0.95 is the confidence coefficient associated with using a particular confidence interval formula, 95% of the time in repeated sampling, confidence intervals calculated using that formula will contain the true value of the population parameter. The flexibility to change confidence coefficients is another benefit of interval estimationcomparedwithpointestimation. Increasing the degree of confidence will widen a confidence interval; decreasing the degree of confidence will narrow a confidence interval.

There are three very common erroneous statements concerning confidence intervals. Here are these misstatements within the realm of using a sample mean, X, to estimate a population mean, m, with 95% confidence.

...