INTERPRETING A T TEST

It is impossible to prove , based on samples, that two population means are exactly equal. What if variable X in the population has a mean of 3.2500 units, and variable Y has a mean of 3.2501 units? Since sample means differ , and the statistical procedures for evaluating differences between means must allow for variability from sample to sample, we will never be able to detect such a small difference in the population. What does happen instead is that we take two samples, compute a t test, and find a large observed significance level. Perhaps we find a probability of .50 that the t value could be observed in a population with no difference. This large observed significance level does not tell us that the means are exactly equal. It just indicates that the results would not be "far out" if the two means are equal in the population. So instead of embracing the null hypothesis and claiming that it is true, we just say that we have no evidence to believe that it is not true. We cannot prove the null hypothesis.