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

SECTION 8.2 | Uncertainty and Errors in Hypothesis Testing 237
the null hypothesis is true. Notice that most of the sample means are near the hypothesized
population mean, μ =15.8, and that means in the critical region are very unlikely to occur.
With an alpha level of α = .05, only 5% of the samples have means in the critical
region. Therefore, there is only a 5% probability (p = .05) that one of these samples will
be obtained. Thus, the alpha level determines the probability of obtaining a sample mean
in the critical region when the null hypothesis is true. In other words, the alpha level determines
the probability of a Type I error.
DEFINITION
The alpha level for a hypothesis test is the probability that the test will lead to a
Type I error. That is, the alpha level determines the probability of obtaining sample
data in the critical region even though the null hypothesis is true.
In summary, whenever the sample data are in the critical region, the appropriate decision
for a hypothesis test is to reject the null hypothesis. Normally this is the correct decision
because the treatment has caused the sample to be different from the original population;
that is, the treatment effect has pushed the sample mean into the critical region. In this case,
the hypothesis test has correctly identified a real treatment effect. Occasionally, however,
sample data are in the critical region just by chance, without any treatment effect. When
this occurs, the researcher will make a Type I error; that is, the researcher will conclude
that a treatment effect exists when in fact it does not. Fortunately, the risk of a Type I error
is small and is under the control of the researcher. Specifically, the probability of a Type I
error is equal to the alpha level.
■ Type II Errors
Whenever a researcher rejects the null hypothesis, there is a risk of a Type I error. Similarly,
whenever a researcher fails to reject the null hypothesis, there is a risk of a Type II
error. By definition, a Type II error is the failure to reject a false null hypothesis. In more
straightforward English, a Type II error means that a treatment effect really exists, but the
hypothesis test fails to detect it.
DEFINITION
A Type II error occurs when a researcher fails to reject a null hypothesis that is
really false. In a typical research situation, a Type II error means that the hypothesis
test has failed to detect a real treatment effect.
A Type II error occurs when the sample mean is not in the critical region even though the
treatment has an effect on the sample. Often this happens when the effect of the treatment
is relatively small. In this case, the treatment does influence the sample, but the magnitude
of the effect is not big enough to move the sample mean into the critical region. Because
the sample is not substantially different from the original population (it is not in the critical
region), the statistical decision is to fail to reject the null hypothesis and to conclude that
there is not enough evidence to say there is a treatment effect.
The consequences of a Type II error are usually not as serious as those of a Type I error.
In general terms, a Type II error means that the research data do not show the results that
the researcher had hoped to obtain. The researcher can accept this outcome and conclude
that the treatment either has no effect or has only a small effect that is not worth pursuing,
or the researcher can repeat the experiment (usually with some improvement, such as a
larger sample) and try to demonstrate that the treatment really does work.