Page 131 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

4 Parametric Tests of Hypotheses

In statistical data analysis an important objective is the capability of making
decisions about population distributions and statistics based on samples. In order to
make such decisions a hypothesis is formulated, e.g. “is one manufacture method
better than another?”, and tested using an appropriate methodology. Tests of
hypotheses are an essential item in many scientific studies. In the present chapter
we describe the most fundamental tests of hypotheses, assuming that the random
variable distributions are known − the so-called parametric tests. We will first,
however, present a few important notions in section 4.1 that apply to parametric
and to non-parametric tests alike.

4.1 Hypothesis Test Procedure

Any hypothesis test procedure starts with the formulation of an interesting
hypothesis concerning the distribution of a certain random variable in the
population. As a result of the test we obtain a decision rule, which allows us to
either reject or accept the hypothesis with a certain probability of error, referred to
as the level of significance of the test.
In order to illustrate the basic steps of the test procedure, let us consider the
following example. Two methods of manufacturing a special type of drill,
respectively A and B, are characterised by the following average lifetime (in
continuous work without failure): µ A = 1100 hours and µ B = 1300 hours. Both
methods have an equal standard deviation of the lifetime, σ = 270 hours. A new
manufacturer of the same type of drills claims that his brand is of a quality
identical to the best one, B, and with lower manufacture costs. In order to assess
this claim, a sample of 12 drills of the new brand were tested and yielded an
average lifetime of x = 1260 hours. The interesting hypothesis to be analysed is
that there is no difference between the new brand and the old brand B. We call it
the null hypothesis and represent it by H 0. Denoting by µ the average lifetime of
the new brand, we then formalise the test as:

H 0: µ =µ B =1300.
H 1: µ =µ A =1100.

Hypothesis H 1 is a so-called alternative hypothesis. There can be many
alternative hypotheses, corresponding to µ ≠µ B. However, for the time being, we
assume that µ =µ A is the only interesting alternative hypothesis. We also assume