Statistical Inference: A Short Course

13.1 Distributional Hypotheses

In Chapter 10, we assumed that a population probability distribution had a particular functional form, for example, we assumed that a continuous random variable X was N(μ, σ). Under this “distributional assumption,” we could then test a parametric hypothesis about, say, the unknown mean μ given that σ is known. For instance, we could test , against using the standard normal distribution. However, if the distributional assumption of normality is not made, then we can actually test the distributional hypothesis is normally distributed, against is not normally distributed.

An important method of testing a distributional hypothesis is Pearson's goodness-of-fit test. Here we are interested in determining if a set of sample observations can be viewed as values of a population random variable having a given probability distribution. So given some distributional hypothesis, this test enables us to determine if the population follows a specific probability model.