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

238 CHAPTER 8 | Introduction to Hypothesis Testing
Unlike a Type I error, it is impossible to determine a single, exact probability for a Type II
error. Instead, the probability of a Type II error depends on a variety of factors and therefore
is a function, rather than a specific number. Nonetheless, the probability of a Type II
error is represented by the symbol β, the Greek letter beta.
In summary, a hypothesis test always leads to one of two decisions.
1. The sample data provide sufficient evidence to reject the null hypothesis and conclude
that the treatment has an effect.
2. The sample data do not provide enough evidence to reject the null hypothesis. In
this case, you fail to reject H 0
and conclude that the treatment does not appear to
have an effect.
In either case, there is a chance that the data are misleading and the decision is wrong. The
complete set of decisions and outcomes is shown in Table 8.1. The risk of an error is especially
important in the case of a Type I error, which can lead to a false report. Fortunately,
the probability of a Type I error is determined by the alpha level, which is completely
under the control of the researcher. At the beginning of a hypothesis test, the researcher
states the hypotheses and selects the alpha level, which immediately determines the risk
of a Type I error.
TABLE 8.1
Possible outcomes of a
statistical decision.
Actual Situation
No Effect
H 0
True
Effect Exists,
H 0
False
Experimenter’s
Decision
Reject H 0
Type I error Decision correct
Fail to Reject H 0
Decision correct Type II error
■ Selecting an Alpha Level
As you have seen, the alpha level for a hypothesis test serves two very important functions.
First, alpha helps determine the boundaries for the critical region by defining the concept
of “very unlikely” outcomes. At the same time, alpha determines the probability of a Type I
error. When you select a value for alpha at the beginning of a hypothesis test, your decision
influences both of these functions.
The primary concern when selecting an alpha level is to minimize the risk of a Type
I error. Thus, alpha levels tend to be very small probability values. By convention, the
largest permissible value is α = .05. When there is no treatment effect, an alpha level of
.05 means that there is still a 5% risk, or a 1-in-20 probability, of rejecting the null hypothesis
and committing a Type I error. Because the consequences of a Type I error can be relatively
serious, many individual researchers and many scientific publications prefer to use
a more conservative alpha level such as .01 or .001 to reduce the risk that a false report is
published and becomes part of the scientific literature. (For more information on the origins
of the .05 level of significance, see the excellent short article by Cowles and Davis, 1982.)
At this point, it may appear that the best strategy for selecting an alpha level is to choose
the smallest possible value to minimize the risk of a Type I error. However, there is a different
kind of risk that develops as the alpha level is lowered. Specifically, a lower alpha level
means less risk of a Type I error, but it also means that the hypothesis test demands more
evidence from the research results.
The trade-off between the risk of a Type I error and the demands of the test is controlled
by the boundaries of the critical region. For the hypothesis test to conclude that the