Confidence Interval

Confidence intervals make a reasoned statement about the true mean of a population based on a random sample. When taking the mean (X) of a random population sample; most likely, the mean of that sample will not be the true mean of the population ( µ ), but rather an estimate. The confidence interval represents a range of values around the sample mean that include the true mean.

Confidence intervals are written with a percentage; what does this percentage represent? If the researcher were to take 100 random samples with a 95% confidence interval for each sample, then he or she expects that for 95 of the 100 samples (95%), the range of values produced by the confidence interval procedure will include the true mean of the population. Although the researcher in practice only has a single sample, the researcher is confident that their interval contains the true mean due to the process used to calculate the confidence interval.

A Random Sample from this Population

The following Java Applet is a visual representation of this notion of confidence. In the Applet there are 20 different data sets, which are separated by the green lines. The blue lines are the data points in the data sets. Each red line represents the confidence range. The small red dash in the center of the red line is the mean and the lines at the top and bottom are the limits of the confidence interval. The pink line is the population mean and is thus measured from 0 because this is a "standard normal" population.

Run the Applet 5 times (which gives you information on 100 data sets) with the alpha level left at its default (.05). Count the number of times the single red line in a data set does not intersect the pink line for every run. After 5 runs, this number should be 5, which gives you your 95% Confidence Interval (100%-5%=95%). For a series of 5 runs, you may get a few more or less than 5 since confidence intervals are based on probability theory. An alpha of .01 is a 99% Confidence Interval (note the larger red lines), which means only 1 line should not intersect in 5 runs (100%-1%=99%), and an alpha of .1 is a 90% Confidence Interval (note the smaller red lines), which means 10 lines should not intersect in 5 runs (100%-10%=90%). Again, remember that because of probability, you may get a few more or less than the target number in any given series of 5 runs.

The width of the confidence interval increases as the confidence level increases, since with a greater width one is more likely to have included the true mean.

In the previous menu, the Baby Weight Activity uses a confidence interval. The activity assumes that you have access to Excel, a TI-83 calculator or another software package capable of performing descriptive tests. To see the actual math used in finding a confidence interval, go to Confidence Interval Math.