Each time the New Sample button is pressed, a pair of new samples from an exponential distribution with population means taken from the sliders are generated. The samples are plotted on a histogram and, underneath, a probability density function is drawn showing the standard error for the sample. As the confidence intervals are changed, observe how the acceptable deviation of the standard error changes as well.

Each time the button is pressed, we could think of it as a significance test with an alpha set to the complement of the confidence interval. In other words, if the probability distributions for the sample means overlap at the 95 percent confidence interval, we cannot reject the null hypothesis at the 5 percent significance level.

Observe how, even when the population means are identical, occasional large deviations in the means will occur. Where samples differ by more than our standard error, we can accept the alternate hypothesis. With a confidence level of...