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

242 CHAPTER 8 | Introduction to Hypothesis Testing
FIGURE 8.7
Sample means that fall in the critical region
(shaded area) have a probability less than
alpha (p < α). In this case, the null hypothesis
should be rejected. Sample means that
do not fall in the critical region have a probability
greater than alpha (p > α).
Reject H 0
p . a
Reject H 0
p , a Fail to reject H 0 p , a
report might state that the treatment effect was significant, with z = 2.25, p = .0244.
When using exact values for p, however, you must still satisfy the traditional criterion for
significance; specifically, the p value must be smaller than .05 to be considered statistically
significant. Remember: The p value is the probability that the result would occur if H 0
were
true (without any treatment effect), which is also the probability of a Type I error. It is
essential that this probability be very small.
■ Factors That Influence a Hypothesis Test
The final decision in a hypothesis test is determined by the value obtained for the z-score
statistic. If the z-score is large enough to be in the critical region, we reject the null hypothesis
and conclude that there is a significant treatment effect. Otherwise, we fail to reject H 0
and conclude that the treatment does not have a significant effect. The most obvious factor
influencing the size of the z-score is the difference between the sample mean and the hypothesized
population mean from H 0
. A big mean difference indicates that the treated sample is
noticeably different from the untreated population and usually supports a conclusion that the
treatment effect is significant. In addition to the mean difference, however, the size of the
z-score is also influenced by the standard error, which is determined by the variability of
the scores (standard deviation or the variance) and the number of scores in the sample (n).
s M
5 s Ïn
Therefore, these two factors also help determine whether the z-score will be large
enough to reject H 0
. In this section we examine how the variability of the scores and the
size of the sample can influence the outcome of a hypothesis test.
We will use the research study from Example 8.1 to examine each of these factors. The
study used a sample of n = 36 male customers and concluded that wearing the color red
has a significant effect on the tips that waitresses receive, z = 2.25, p < .05.
The Variability of the Scores In Chapter 4 (page 124), we noted that high variability can
make it very difficult to see any clear patterns in the results from a research study. In a hypothesis
test, higher variability can reduce the chances of finding a significant treatment effect.
For the study in Example 8.1, the standard deviation is σ = 2.4. With a sample of n = 36,
this produced a standard error of σ M
= 0.4 points and a significant z-score of z = 2.25.
Now consider what happens if the standard deviation is increased to σ = 5.4. With the
increased variability, the standard error becomes σ M
= 5.4 = 0.9 points. Using the same
Ï36
0.9-point mean difference from the original example the new z-score becomes
z 5 M 2m 16.7 2 15.8
5 5 0.9
s M
0.9 0.9 5 1.00