Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

Hypothesis Testing

Although we normally base our decisions on knowledge about the problems that we are trying to solve, we always have to accept some risk of making an error. Our knowledge is never perfect and complete. Depending on the consequences of the decision, we require different degrees of confidence that the decision is correct. Non-critical questions such as whether to take an umbrella with us when we leave for work, need only low levels of confidence. Whilst in contrast, critical decisions like weighting the evidence in a murder case in court, or diagnosing a certain disease in a hospital, require a high degree of confidence. The uncertainty in the process of making a decision is related to the fact that the probability of making an error is not zero.

Hypothesis testing gives us the guidelines for choosing between alternatives by either controlling or minimizing the error associated with the decision. The simplest case for a decision is the 'yes-or-no' question. In court, for example, the jurors have to decide "guilty or not guilty". These statements are two hypotheses. The normal assumption is "not guilty", in statistics this is called the null hypothesis. It is what we normally assume. Then there is an alternative hypothesis, in our example "guilty". We will accept this alternative hypothesis only when there is convincing evidence.

Hypothesis testing can be summarized in the questions: is it reasonable to assume that the value of a population parameter is equal to / larger than / less than x? This question can be applied in various situations. The population parameter can be either the mean or the variance. The value of x is either specified on the basis of prior knowledge, or an estimated parameter from another population.

Hypothesis testing is always a five-step procedure:

  • Formulation of the null and the alternative hypotheses
  • Specification of the level of significance
  • Calculation of the test statistic
  • Definition of the region of rejection
  • Selection of the appropriate hypothesis