Applied Statistics

3Testing Hypotheses – One‐ and Two‐Sample Problems

3.1 Introduction

In empirical research, a scientist often formulates conjectures about objects of his research. For instance, he may argue that the fat content in the milk of Jersey cows is higher than that of Holstein Friesians. To check conjectures, he will perform an experiment. Now statistics come into play.

Sometimes the aim of investigation is not to determine certain statistics (to estimate parameters), but to test or to examine carefully considered hypotheses (assumptions, suppositions) and often also wishful notions based on practical material. In addition, in this case we establish a mathematical model where the hypothesis is formulated in the form of model parameters.

We assume that we have one random sample or two random samples from special distributions. We begin with the one‐sample problem and assume that the distribution of the components of the sample depends on a parameter (vector) θ. We would like to test a hypothesis about θ. First, we define what we have to understand by these terms.

A statistical test is a procedure that allows a decision for accepting or rejecting a hypothesis about the unknown parameter to occur in the distribution of a random variable. We shall suppose in the following that two hypotheses are possible. The first (or main) hypothesis is the null hypothesis H₀, the other one is the alternative hypothesis H_A. The hypothesis H₀ is right, if H_A is wrong, and vice versa. Hypotheses can be composite ...