5.1 BASIC CONCEPTS
From the introduction of sampling distributions in Chapter 4, it was clear that
the value of a sample mean is influenced by:
- The populationm, because
mx¼m- Chance;xandmare almost never identical. The variance of the sampling
distribution is
s^2 x¼
s^2
na combined e¤ect of natural variation in the populationðs^2 Þand sample
sizen.Therefore, when an observed valuexis far from a hypothesized value ofm
(e.g., mean high blood pressures for a group of oral contraceptive users com-
pared to a typical average for women in the same age group), a natural ques-
tion would be: Was it just due to chance, or something else? To deal with
questions such as this, statisticians have invented the concept ofhypothesis
tests, and these tests have become widely used statistical techniques in the
health sciences. In fact, it is almost impossible to read a research article in
public health or medical sciences without running across hypothesis tests!
5.1.1 Hypothesis Tests
When a health investigator seeks to understand or explain something, for
example the e¤ect of a toxin or a drug, he or she usually formulates his or
her research question in the form of ahypothesis. In the statistical context, a
hypothesis is a statement about a distribution (e.g., ‘‘the distribution is nor-
mal’’) or its underlying parameter(s) (e.g., ‘‘m¼10’’), or a statement about the
relationship between probability distributions (e.g., ‘‘there is no statistical rela-
tionship’’) or its parameters (e.g., ‘‘m 1 ¼m 2 ’’—equality of population means).
The hypothesis to be tested is called thenull hypothesisand will be denoted by
H 0 ; it is usually stated in the null form, indicating no di¤erence or no relation-
ship between distributions or parameters, similar to the constitutional guaran-
tee that the accused is presumed innocent until proven guilty. In other words,
under the null hypothesis, an observed di¤erence (like the one between sample
meansx 1 andx 2 for sample 1 and sample 2, respectively) just reflects chance
variation. Ahypothesis testis a decision-making process that examines a set or
sets of data, and on the basis of expectation underH 0 , leads to a decision as to
whether or not to rejectH 0 .Analternative hypothesis, which we denote byHA,
190 INTRODUCTION TO STATISTICAL TESTS OF SIGNIFICANCE