 # Confidence Interval

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Definition

A confidence interval for a given population parameter gives an estimated range of values which includes the true value of the parameter, based upon a pre-assigned probability called the confidence level. It provides more information about the parameter of interest than can be obtained by simply performing a hypothesis test. Hence, it is often used when the investigator knows that an effect exists but they want to estimate the possible range of values for the parameter with a level of certainty.

Examples

Consider a 100(1-α)% confidence interval for the population mean:

L ≤ µ ≤ U

where L is the lower confidence limit and U is the upper confidence limit. Interpretation: if a large number of such intervals are constructed in repeated random samplings of the same size, then 100(1-α)% of the intervals will contain the true parameter value. Here (1-α) is the confidence level/coefficient.

Application

Whenever the confidence interval is constructed as an inversion of the corresponding hypothesis test, it can be used to decide the test at a significance level that equals (1-the confidence level) of the interval. In such cases the interval consists of the set of values of the parameter for which we would fail to reject the null hypothesis.

When the test is one-sided (less-than or greater-than) or when the parameter is expected to take on values in one direction only, then a lower (or upper) confidence bound is calculated which gives the value above (or below) which the true parameter value lies.