is a hypothesis that in some sense contradicts the null hypothesisH 0 , just as the
chargeby the prosecution in a trial by jury. UnderHA, the observed di¤erence
is real (e.g.,x 10 x 2 not by chance but becausem 10 m 2 ). A null hypothesis is
rejected if and only if there is su‰ciently strong evidence from the data to sup-
port its alternative—the names are somewhat unsettling, because the alterna-
tive hypothesis is, for a health investigator, the one that he or she usually wants
to prove. (The null hypothesis is just a dull explanation of the findings–in terms
of chance variation!) However, these are entrenched statistical terms and will
be used as standard terms for the rest of this book.
Why is hypothesis testing important? Because in many circumstances we
merely wish to know whether a certain proposition is true or false. The process
of hypothesis tests provides a framework for making decisions on anobjective
basis, by weighing the relative merits of di¤erent hypotheses, rather than on a
subjectivebasis by simply looking at the numbers. Di¤erent people can form
di¤erent opinions by looking at data (confounded by chance variation or sam-
pling errors), but a hypothesis test provides a standardized decision-making
process that will be consistent for all people. The mechanics of the tests vary
with the hypotheses and measurement scales (Chapters 6, 7, and 8), but the
general philosophy and foundation is common and is discussed in some detail
in this chapter.
5.1.2 Statistical Evidence
A null hypothesis is often concerned with a parameter or parameters of pop-
ulation(s). However, it is often either impossible, or too costly or time con-
suming, to obtain the entire population data on any variable in order to see
whether or not a null hypothesis is true. Decisions are thus made using sample
data. Sample data are summarized into a statistic or statistics that are used to
estimate the parameter(s) involved in the null hypothesis. For example, if a null
hypothesis is aboutm(e.g.,H 0 :m¼10), a good place to look for information
aboutmisx. In that context, the statisticxis called atest statistic; a test sta-
tistic can be used to measure the di¤erence between the data (i.e., the numerical
value ofxobtained from the sample) and what is expected if the null hypothe-
sis is true (i.e., ‘‘m¼10’’). However, this evidence is statistical evidence; it
varies from sample to sample (in the context of repeated sampling). It is a
variable with a specific sampling distribution. The observed value is thus usu-
ally converted to a standard unit: the number of standard errors away from a
hypothesized value. At this point, the logic of the test can be seen more clearly.
It is an argument by contradiction, designed to show that the null hypothesis
will lead to a less acceptable conclusion (an almost impossible event—some
event that occurs with near-zero probability) and must therefore be rejected. In
other words, the di¤erence between the data and what is expected on the null
hypothesis would be very di‰cult—even absurd—to explain as a chance vari-
ation; it makes you want to abandon (or reject) the null hypothesis and believe
in the alternative hypothesis because it is more plausible.
BASIC CONCEPTS 191