Fortunately, even though the conditions for a one-sample [latex]t[/latex] confidence interval are not met, there is another method that can be used to get a confidence interval for the population mean. This new type of interval is called a bootstrap confidence interval.

Bootstrap Distributions

The [latex]t[/latex] confidence interval is constructed by creating an interval around the observed sample mean by adding and subtracting a number to get the endpoints of the confidence interval. The number that is added and subtracted depends on certain conditions being met and is based on the standard deviation of the sample mean and the [latex]t[/latex] Distribution. Bootstrapping is a way to figure out what number should be added to the sample mean and what number should be subtracted from the sample mean if we can’t rely on the [latex]t[/latex] Distribution.

What we do is sample from a hypothetical population that we think will be very similar to the population which our sample is from. Seeing what happens when we sample from this hypothetical population gives us the information that we need to determine the endpoints of a reasonable confidence interval.

If we assume that our sample is representative of the population, the hypothetical population that we would be thinking of would be one that has the same values as our sample, but that is much larger than the sample. For example, if one of the observations in our sample was [latex]30[/latex], we would think that there were probably many individuals in the population that had a value of [latex]30[/latex]. This is why, when we create a bootstrap sample, we sample with replacement. This is equivalent to sampling from the larger hypothetical population that we think is similar to the population we are actually interested in.