Hypothesis testing is a formal scientific process that accounts for statistical uncertainty.

**Tests of Hypotheses**

**TERMINOLOGY**

Hypothesis testing is a formal scientific process
that accounts for statistical uncertainty. As such, the process involves much
new statistical terminology that we now introduce. A hypothesis is a statement
of belief about the values of population para-meters. In hypothesis testing, we
usually consider two hypotheses: the null and al-ternative hypotheses. The null
hypothesis, denoted by *H*_{0},
is usually a hypothesis of no difference. Initially, we will consider a type of
*H*_{0} that is a claim that
there is no difference between the population parameter and its hypothesized
value or set of values. The hypothesized values chosen for the null hypothesis
are usually chosen to be uninteresting values. An example might be that in a
trial comparing two dia-betes drugs, the mean values for fasting plasma glucose
are the same for the two treatment groups.

In general, the experimenter is interested in
rejecting the null hypothesis. The al-ternative hypothesis, denoted by *H*_{1}, is a claim that the null
hypothesis is false; i.e., the population parameter takes on a value different
from the value or values specified by the null hypothesis. The alternative
hypothesis is usually the scientifically inter-esting hypothesis that we would
like to confirm. By using probability theory, our goal is to lend credence to
the alternative hypothesis by rejecting the null hypothesis. In the diabetes
example, an interesting alternative might be that the fasting plasma glu-cose
mean is significantly (both statistically and clinically) lower for patients
with the experimental drug as compared to the mean for patients with the
control drug.

Because of statistical uncertainty regarding
inferences about population parame-ters based on sample data, we cannot prove
or disprove either the null or the alter-native hypotheses. Rather, we make a
decision based on probability and accept a probability of making an incorrect
decision.

The type I error is defined as the probability of
falsely rejecting the null hypoth-esis; i.e., to claim on the basis of data
from a sample that the true parameter is not a value specified by the null
hypothesis when in fact it is. In other words, a type I er-ror occurs when the
null hypothesis is true but we incorrectly reject *H*_{0}. The other possible mistake we can make is to not
reject the null hypothesis when the true pa-rameter value is specified by the
alternative hypothesis. This kind of error is called a type II error.

Based on the observed data, we form a statistic
(called a test statistic) and con-sider its sampling distribution in order to
define critical values for rejecting the null hypothesis. For example, the *Z* and *t* statistics covered previously (refer to Chapter 8) can serve as
test statistics for those population parameters. A statistician uses one or
more cutoff values for the test statistic to determine when to reject or not to
reject the null hypothesis.

These cutoff values are called critical values; the
set of values for which the null hypothesis would be rejected is called the
critical region, or rejection region. The other values of the test statistic
form a region that we will call the nonrejection re-gion. We are tempted to
call the nonrejection region the acceptance region; howev-er, we hesitate to do
so because the Neyman–Pearson approach to hypothesis test-ing chooses the
critical value to control the type I error, but the type II error then depends
on the specific value of the parameter when the alternative is true. In the
next section, we will discuss this point in detail as well as the
Neyman–Pearson ap-proach.

The probability of observing a value in the
critical region when the null hypothe-sis is correct is called the significance
level; the hypothesis test is also called a test of significance. The
significance level is denoted by α, which
often is set at a low value such as 0.01 or 0.05. These values also can be
termed error levels; i.e., we are acknowledging that it is acceptable to be
wrong one time out of 100 tests or five times out of 100 tests, respectively.
The symbol is also the probability of a type I error; the symbol β is used to denote the probability of a type II error, as explained in
Section 9.7.

Given a test statistic and an observed value, one
can compute the probability of observing a value as extreme or more extreme
than the observed value when the null hypothesis is true. This probability is
called the *p*-value. The *p*-value is related to the significance
level in that if we had chosen the critical value to be equal to the observed
value of the test statistic, the *p*-value
would be equal to the significance level.

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