Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

230 CHAPTER 8 | Introduction to Hypothesis Testing
The distribution of sample means
if the null hypothesis is true
(all the possible outcomes)
FIGURE 8.4
The set of potential samples
is divided into those that are
likely to be obtained and
those that are very unlikely
to be obtained if the null
hypothesis is true.
Extreme, lowprobability
values
if H 0 is true
Sample means
close to H 0 :
high-probability values
if H 0 is true
m from H 0
Extreme, lowprobability
values
if H 0 is true
The extremely unlikely values, as defined by the alpha level, make up what is called the
critical region. These extreme values in the tails of the distribution define outcomes that
are not consistent with the null hypothesis; that is, they are very unlikely to occur if the null
hypothesis is true. Whenever the data from a research study produce a sample mean that
is located in the critical region, we conclude that the data are not consistent with the null
hypothesis, and we reject the null hypothesis.
DEFINITIONS
The alpha level, or the level of significance, is a probability value that is used to
define the concept of “very unlikely” in a hypothesis test.
The critical region is composed of the extreme sample values that are very unlikely
(as defined by the alpha level) to be obtained if the null hypothesis is true. The
boundaries for the critical region are determined by the alpha level. If sample data
fall in the critical region, the null hypothesis is rejected.
Technically, the critical region is defined by sample outcomes that are very unlikely to
occur if the treatment has no effect (that is, if the null hypothesis is true). Reversing the
point of view, we can also define the critical region as sample values that provide convincing
evidence that the treatment really does have an effect. For our example, the regular
population of male customers leaves a mean tip of μ = 15.8 percent. We selected a sample
from this population and administered a treatment (the red shirt) to the individuals in the
sample. What kind of sample mean would convince you that the treatment has an effect? It
should be obvious that the most convincing evidence would be a sample mean that is really
different from μ = 15.8 percent. In a hypothesis test, the critical region is determined by
sample values that are “really different” from the original population.
The Boundaries for the Critical Region To determine the exact location for the
boundaries that define the critical region, we use the alpha-level probability and the unit