Confidence Interval for a Population Mean

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This Demonstration illustrates the interpretation of a 95% confidence for a population mean. Assume that the population is described by a normal distribution with a known population standard deviation of . Also assume that the population mean is . The confidence interval is based on a sample size of .


The formula for the confidence interval is [1]. Many textbooks give an interpretation by saying we have 95% confidence that the population mean is surrounded by the endpoints of the observed confidence interval [2]. Other books say that we should consider replicating the experiment a very large number of times. We expect that 95% of the samples will produce a confidence interval whose endpoints surround the population mean [1].

A much more direct interpretation is that either the observed confidence interval surrounds the population mean (of in this example) or we got an unusual set of data. This interpretation is adopted in this Demonstration. The graphic displays the probability density of the random sample mean based on a normal distribution with , and a sample size of . The shaded area defines an interval of values of with a probability of .. We call these typical values of . Such a value of also corresponds to a typical set of data. The black horizontal bar has endpoints that are the endpoints of a possible 95% confidence interval. The dot in the center bar is a possible sample mean. If the dot is in the shaded area (corresponding to a typical dataset), then the endpoints of the confidence interval will surround the population mean of . If a deviant value of (outside of the shaded region) is observed, the endpoints of the confidence interval will not be symmetric about .


Contributed by: Peyton Cook (August 2022)
Open content licensed under CC BY-NC-SA




[1] J. Devore, Probability and Statistics for Engineering and the Sciences, 9th ed., Boston: Cengage Learning, 2016.

[2] W. Navidi, Principles of Statistics for Engineers and Scientists, Dubuque, IA: McGraw-Hill, 2010.

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