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

SECTION 8.4 | Directional (One-Tailed) Hypothesis Tests 247
If the prediction is
that the treatment will
produce a decrease
in scores, the critical
region is located entirely
in the left-hand tail of
the distribution.
■ The Critical Region for Directional Tests
The critical region is defined by sample outcomes that are very unlikely to occur if
the null hypothesis is true (that is, if the treatment has no effect). When the critical
region was first defined (page 232), we noted that the critical region can also be defined
in terms of sample values that provide convincing evidence that the treatment really
does have an effect. For a directional test, the concept of “convincing evidence” is
the simplest way to determine the location of the critical region. We begin with all the
possible sample means that could be obtained if the null hypothesis is true. This is
the distribution of sample means and it will be normal (because the population of test
scores is normal), have an expected value of μ = 15.8 (from H 0
), and, for a sample of
n = 36, will have a standard error of σ M
= 2.4 = 0.4 points. The distribution is shown in
Ï36
Figure 8.8.
For this example, the treatment is expected to increase test scores. If the regular population
of male customers has an average tip of μ = 15.8 percent, then a sample mean that is
substantially more than 15.8 would provide convincing evidence that the red shirt worked.
Thus, the critical region is located entirely in the right-hand tail of the distribution corresponding
to sample means much greater than μ = 15.8 (Figure 8.8). Because the critical
region is contained in one tail of the distribution, a directional test is commonly called a
one-tailed test. Also note that the proportion specified by the alpha level is not divided
between two tails, but rather is contained entirely in one tail. Using α = .05 for example,
the whole 5% is located in one tail. In this case, the z-score boundary for the critical region
is z = 1.65, which is obtained by looking up a proportion of .05 in column C (the tail) of
the unit normal table.
Notice that a directional (one-tailed) test requires two changes in the step-by-step
hypothesis-testing procedure.
1. In the first step of the hypothesis test, the directional prediction is incorporated into
the statement of the hypotheses.
2. In the second step of the process, the critical region is located entirely in one tail of
the distribution.
After these two changes, a one-tailed test continues exactly the same as a regular two-tailed
test. Specifically, you calculate the z-score statistic and then make a decision about H 0
depending on whether the z-score is in the critical region.
s M
5 0.4
Reject H 0
Data indicate
that H 0 is wrong
FIGURE 8.8
Critical region for Example 8.3.
15.8
m
0 1.65
M
z