Statistical Inference: A Short Course

10.1 What is a Statistical Hypothesis?

In general, a hypothesis is essentially an unproved theory or assertion; it is tentatively accepted as an explanation of certain facts or observations. A statistical hypothesis is a testable hypothesis—it specifies a value for some parameter (call it θ) of a theoretical population distribution. For instance, θ might be a mean or variance or proportion of successes in a population. (This is in contrast to what is called a maintained hypothesis—an assumption that we are willing to believe in and which is thought to hold at least approximately, for example, the population is normally distributed.)

To develop a test procedure, we must actually specify a “pair of hypotheses” in order to admit alternative possibilities for a population. The first hypothesis in the pair is the null hypothesis (denoted H₀)—it is the hypothesis to be tested and either rejected or not rejected. The null hypothesis is “always assumed to be true.” (Incidentally, the word “null” is to be interpreted in the context of “no difference” between the true value of θ and its hypothesized value.) The null hypothesis may be either simple (θ is hypothesized to equal a single numerical value) or compound (θ is hypothesized to fall within a range of values.) The second hypothesis in the pair is the alternative hypothesis (denoted as H₁)—it is the hypothesis that states what the population would look like if the null hypothesis were untrue.

For instance, three typical cases or pairs ...