# Confidence Intervals for single sample t-tests

Whenever we reject the null hypothesis we are saying that the sample we are dealing with (
) does not come from the same population as the one we for which we know the paramaters (μ
). If the distance between the sample mean and the population mean is great enough, we must conclude that the sample mean is part of some other population. This other population has it's own set of parameters (i.e. it's own mean and standard deviation). The sample mean we have is a sample from this other population that we don't know the parameters for yet. Recall, however, that a sample mean is going to be our best estimate of the population mean. When we calculate a confidence interval, we are using the sample mean to estimate the parameters of this new population.
In the following example we reject the null hypothesis that students who cram for an exam score as well as students that do not cram. The sample mean for the crammers was 70 with a standard deviation of 10.3. The sample size was N=40. We rejected the null hypothesis that this sample of crammers did as well on an exam as non-crammers who had a population mean of 75 (Alpha=.05; two-tail). In the example we demonstrate how to calculate the 95% confidence interval. We will be 95% sure that the mean of the new population (from which the sample comes) falls within the range we calculate.