You may marvel at the roundabout procedure whereby a researcher tests a null hypothesis that is believed to be untrue in the hope of rejecting it and thereby accepting the alternative hypothesis that is believed to be true. On reflection, you may recall a similar procedure taught in plane geometry and algebra the method of indirect proof. This method consists of listing all possible answers or solutions to a problem and showing that all but one is contrary to known fact or lead to an absurdity. By a process of elimination, the one that is not contrary to known fact or absurd must be true. The success of the method of indirect proof depends on listing all possibilities and finding a contradiction for all but one. The comparable procedure in testing a null hypothesis consists of formulating the null and alternative hypotheses so that they exhaust all the possibilities concerning a population parameter.
A sample is obtained from the population, and appropriate statistics, such as the sample mean and standard deviation, are computed. If it is highly improbable that the obtained value of the sample mean would have occurred if the null hypothesis were true, then the null hypothesis must be considered a poor prediction of the population mean and should be rejected in favor of the alternative hypothesis.
There is one important difference between the method of indirect proof and null hypothesis testing. In indirect proof, a possibility is rejected only if it is found to lead to a contradiction to known fact or is absurd. In hypothesis testing, the null hypothesis is rejected if they obtained value of a sample mean is very unlikely if the null hypothesis is indeed true. It follows that null hypothesis testing, unlike the method of indirect proof, does not provide incontrovertible proof because the null hypothesis is rejected because of the occurrence of an event that is improbable but not impossible.
If the null hypothesis is not rejected, what conclusion can the researcher draw? Is the null hypothesis true? Not necessarily; there are always alternative reasons for why the null hypothesis is not rejected.
The null hypothesis is true and should not be rejected.
The null hypothesis is false and should be rejected, but the particular sample that was used to estimate m and s is not representative of the population.
The null hypothesis is false and should be rejected, but the experimental methodology is not sufficiently sensitive to detect the true situation.
An experimental methodology can lack sensitivity for a variety of reasons: the size of the sample is too small; the procedure used to measure the dependent variable is subject to large random or systematic errors, and so on.
