LibraryPirate

7.1 INTRODUCTION 221
8. Statistical decision. The statistical decision consists of rejecting or of not rejecting
the null hypothesis. It is rejected if the computed value of the test statistic falls
in the rejection region, and it is not rejected if the computed value of the test statistic
falls in the nonrejection region.
9. Conclusion. If H 0 is rejected, we conclude that H A is true. If H 0 is not rejected,
we conclude that H 0 may be true.
10. p values. The p value is a number that tells us how unusual our sample results
are, given that the null hypothesis is true. A p value indicating that the sample
results are not likely to have occurred, if the null hypothesis is true, provides justification
for doubting the truth of the null hypothesis.
DEFINITION
A p value is the probability that the computed value of a test statistic is
at least as extreme as a specified value of the test statistic when the null
hypothesis is true. Thus, the p value is the smallest value of A for which
we can reject a null hypothesis.
We emphasize that when the null hypothesis is not rejected one should not say that
the null hypothesis is accepted. We should say that the null hypothesis is “not rejected.”
We avoid using the word “accept” in this case because we may have committed a type II
error. Since, frequently, the probability of committing a type II error can be quite high, we
do not wish to commit ourselves to accepting the null hypothesis.
Figure 7.1.2 is a flowchart of the steps that we follow when we perform a hypothesis
test.
Purpose of Hypothesis Testing The purpose of hypothesis testing is to
assist administrators and clinicians in making decisions. The administrative or clinical
decision usually depends on the statistical decision. If the null hypothesis is rejected, the
administrative or clinical decision usually reflects this, in that the decision is compatible
with the alternative hypothesis. The reverse is usually true if the null hypothesis is not
rejected. The administrative or clinical decision, however, may take other forms, such as
a decision to gather more data.
We also emphasize that the hypothesis testing procedures highlighted in the remainder
of this chapter generally examine the case of normally distributed data or cases where
the procedures are appropriate because the central limit theorem applies. In practice, it
is not uncommon for samples to be small relative to the size of the population, or to
have samples that are highly skewed, and hence the assumption of normality is violated.
Methods to handle this situation, that is distribution-free or nonparametric methods, are
examined in detail in Chapter 13. Most computer packages include an analytical procedure
(for example, the Shapiro-Wilk or Anderson-Darling test) for testing normality. It
is important that such tests are carried out prior to analysis of data. Further, when testing
two samples, there is an implicit assumption that the variances are equal. Tests for
this assumption are provided in Section 7.8. Finally, it should be noted that hypothesis