Today, let’s explore some common examples of hypotheses to contrast with the conjectures already discussed.

In defining “hypothesis,” the ability to be tested was a critical element. Making predictions and testing hypotheses is an essential part of statistical inference. To formulate such tests, usually some conjecture has been put forward, either because it is believed to be true or because it is to be used as a basis for argument, but has not been proven. Example: claiming a new drug is more effective than the current drug in alleviating symptoms from some specified disease. How does one go about proving or disproving such a statement?

The common way to test a hypothesis such as that one is similar to the method used to solve some of the puzzles presented here (hmm!). That is, instead of looking at the hypothesis itself (often called “alternative hypothesis”), we look at the “null hypothesis.” The null hypothesis for the example of a new drug, A, which is thought to be better than the current drug, B, would be “The effects of drug A are not statistically differentiable from the effects of drug B.”

So a hypothesis is generally examined by looking first at the null hypothesis. In statistical reports, one often finds statements such as “the null hypothesis is rejected in favor of the alternative hypothesis” or “the null hypothesis is not rejected by the tests.” Of course in deciding whether or not to accept the null hypothesis, one has all the issues of decision theory that we have presented in the past such a type I and type II errors. Type I errors occur when we falsely reject a null hypothesis that is true. Type II errors occur when the null hypothesis is not rejected when it is false. Experimenters usually have some freedom to trade off the probability of making either type of error by how they set up the decision criteria. For instance, in considering the hypothesis that a suspect is an armed suicide bomber, one would prefer to err on the side of making false accusations rather than err on missing a real bomber. The way to make this tradeoff rationally is described in my tutorial, but is beyond the scope of this article.

Note that the reason for considering the null hypothesis for testing rather than the alternative hypothesis is that the null hypothesis relates to the statement (hypothesis) being tested, whereas the alternative hypothesis relates to the statement to be accepted if the null hypothesis is rejected. Concluding “Do not reject the null hypothesis” does not necessarily mean that the null hypothesis is true, it only means we do not have sufficient evidence to reject the null in favor of the alternative. In this roundabout way, rejecting the null hypothesis suggests that the alternative hypothesis is true to some degree of confidence. Details about how to compute the degree of confidence are also beyond the scope of this article, but can be found in any introductory text on statistics.

While all this sounds rather complicated and uninteresting, it only formalizes the processes we often use in solving problems. In the context of puzzles, we often solve them by assuming the null version and then either rejecting it or not.

In distinguishing between conjectures and hypotheses, note that this level of serious distinction between the conjecture and its null is not an issue. It’s only when we come to set up a formalism to test a statement that we find we need a way to proceed logically. Anyone can put out a conjecture, but testing a hypothesis takes some work and has statistics built into the answer.

In the next article, we will look at some examples of true theories and compare them to the conjectures that are often falsely called theories.

For those who wish to delve further into decision theory without wading through a lot of equations, I have posted a tutorial on elementary decision theory. It shows examples of faulty physicians’ diagnoses (important for those considering surgery) and how to evaluate anti-terrorist activities (important for everyone). That tutorial can be found here.