Confidence Interval

A confidence interval is a range of values around an expected outcome within which the actual outcome is expected to fall a specified percentage of the time. In other words, it is a range for which one can assert with a given probability that it will contain a parameter it intends to estimate.

Construction of a Confidence Interval

where

• Point estimate is an estimate of the parameter
• Reliability factor is a number based on the assumed distribution of the point estimate and degree of confidence for the confidence interval
• Standard error is the standard error of sample statistic providing the point estimate

Now let's say we wanted to construct a confidence interval for the population mean when we are sampling from a normal distribution with a known variance. We would use the z-distribution and thus the following equation.

The first variable is our point estimate, the sample mean. The second is a reliability factor. This can be thought of as the z-score which leaves α/2 of probability in the upper right tail. The known standard deviation of the population divided by the square root of the sample size provides the standard error.

Commonly Used Reliability Factors For Constructing Confidence Intervals

z_{α/2 = 1.645 for 90% confidence intervals}
z_{α/2 = 1.960 for 95% confidence intervals}
z_{α/2 = 2.575 for 99% confidence intervals}