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

232 CHAPTER 8 | Introduction to Hypothesis Testing
hypothesis were true—that is, the entire set of sample means that could be obtained if the
treatment has no effect (see Figure 8.5). Now we calculate a z-score that identifies where
our sample mean is located in this hypothesized distribution. The z-score formula for a
sample mean is
z 5 M 2m
s M
In the formula, the value of the sample mean (M) is obtained from the sample data, and the
value of μ is obtained from the null hypothesis. Thus, the z-score formula can be expressed
in words as follows:
sample mean 2 hypothesized population mean
z 5
standard error between M and m
Notice that the top of the z-score formula measures how much difference there is between
the data and the hypothesis. The bottom of the formula measures the standard distance that
ought to exist between a sample mean and the population mean.
STEP 4
Make a decision. In the final step, the researcher uses the z-score value obtained in Step 3
to make a decision about the null hypothesis according to the criteria established in Step 2.
There are two possible outcomes.
1. The sample data are located in the critical region. By definition, a sample value in
the critical region is very unlikely to occur if the null hypothesis is true. Therefore,
we conclude that the sample is not consistent with H 0
and our decision is to reject
the null hypothesis. Remember, the null hypothesis states that there is no treatment
effect, so rejecting H 0
means we are concluding that the treatment did have an
effect.
For the example we have been considering, suppose the sample produced a mean
tip of M = 16.7 percent. The null hypothesis states that the population mean is
μ = 15.8 percent and, with n = 36 and σ = 2.4, the standard error for the sample mean is
s M
5 s Ïn 5 2.4
6 5 0.4
Thus, a sample mean of M = 16.7 produces a z-score of
z 5 M 2m 16.7 2 15.8
5 5 0.9
s M
0.4 0.4 5 2.25
With an alpha level of α = .05, this z-score is far beyond the boundary of 1.96.
Because the sample z-score is in the critical region, we reject the null hypothesis and
conclude that the red shirt did have an effect on tipping behavior.
2. The sample data are not in the critical region. In this case, the sample mean is
reasonably close to the population mean specified in the null hypothesis (in the
center of the distribution). Because the data do not provide strong evidence that the
null hypothesis is wrong, our conclusion is to fail to reject the null hypothesis. This
conclusion means that the treatment does not appear to have an effect.
For the research study examining the effect of a red shirt, suppose our sample
produced a mean tip of M = 16.1 percent. As before, the standard error for a sample
of n = 36 is σ M
= 0.4, and the null hypothesis states that μ = 15.8 percent. These
values produce a z-score of
z 5 M 2m 16.1 2 15.8
5 5 0.3
s M
0.4 0.4 5 0.75