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

SECTION 8.1 | The Logic of Hypothesis Testing 229
Notice that the alternative hypothesis simply states that there will be some type of change.
It does not specify whether the effect will be increased or decreased tips. In some circumstances,
it is appropriate for the alternative hypothesis to specify the direction of
the effect. For example, the researcher might hypothesize that a red shirt will increase
tips (μ > 15.8). This type of hypothesis results in a directional hypothesis test, which is
examined in detail later in this chapter. For now we concentrate on nondirectional tests,
for which the hypotheses simply state that the treatment has no effect (H 0
) or has some
effect (H 1
).
STEP 2
Set the criteria for a decision. Eventually the researcher will use the data from the
sample to evaluate the credibility of the null hypothesis. The data will either provide support
for the null hypothesis or tend to refute the null hypothesis. In particular, if there is a
big discrepancy between the data and the hypothesis, we will conclude that the hypothesis
is wrong.
To formalize the decision process, we use the null hypothesis to predict the kind of
sample mean that ought to be obtained. Specifically, we determine exactly which sample
means are consistent with the null hypothesis and which sample means are at odds with
the null hypothesis.
For our example, the null hypothesis states that the red shirts have no effect and the
population mean is still μ = 15.8 percent, the same as the population mean for waitresses
wearing white shirts. If this is true, then the sample mean should have a value around 15.8.
Therefore, a sample mean near 15.8 is consistent with the null hypothesis. On the other
hand, a sample mean that is very different from 15.8 is not consistent with the null hypothesis.
To determine exactly which values are “near” 15.8 and which values are “very different
from” 15.8, we will examine all of the possible sample means that could be obtained
if the null hypothesis is true. For our example, this is the distribution of sample means for
n = 36. According to the null hypothesis, this distribution is centered at μ = 15.8. The
distribution of sample means is then divided into two sections.
1. Sample means that are likely to be obtained if H 0
is true; that is, sample means that
are close to the null hypothesis
2. Sample means that are very unlikely to be obtained if H 0
is true; that is, sample
means that are very different from the null hypothesis
Figure 8.4 shows the distribution of sample means divided into these two sections.
Notice that the high-probability samples are located in the center of the distribution and
have sample means close to the value specified in the null hypothesis. On the other hand,
the low-probability samples are located in the extreme tails of the distribution. After the
distribution has been divided in this way, we can compare our sample data with the values
in the distribution. Specifically, we can determine whether our sample mean is consistent
with the null hypothesis (like the values in the center of the distribution) or whether our
sample mean is very different from the null hypothesis (like the values in the extreme tails).
With rare exceptions,
an alpha level is never
larger than .05.
The Alpha Level To find the boundaries that separate the high-probability samples
from the low-probability samples, we must define exactly what is meant by “low” probability
and “high” probability. This is accomplished by selecting a specific probability
value, which is known as the level of significance, or the alpha level, for the hypothesis
test. The alpha (α) value is a small probability that is used to identify the low-probability
samples. By convention, commonly used alpha levels are α = .05 (5%), α = .01 (1%),
and α = .001 (0.1%). For example, with α = .05, we separate the most unlikely 5% of the
sample means (the extreme values) from the most likely 95% of the sample means (the
central values).