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9-1 HYPOTHESIS TESTING 279

Since we use probability distributions to represent populations, a statistical hypothesis
may also be thought of as a statement about the probability distribution of a random variable.
The hypothesis will usually involve one or more parameters of this distribution.
For example, suppose that we are interested in the burning rate of a solid propellant used
to power aircrew escape systems. Now burning rate is a random variable that can be described
by a probability distribution. Suppose that our interest focuses on the mean burning rate (a
parameter of this distribution). Specifically, we are interested in deciding whether or not the
mean burning rate is 50 centimeters per second. We may express this formally as

H : 50 centimeters per second
0
H : 50 centimeters per second (9-1)
1
The statement H 0 : 50 centimeters per second in Equation 9-1 is called the null
hypothesis, and the statement H : 50 centimeters per second is called the alternative
1

hypothesis. Since the alternative hypothesis specifies values of that could be either greater
or less than 50 centimeters per second, it is called a two-sided alternative hypothesis. In some
situations, we may wish to formulate a one-sided alternative hypothesis, as in
H : 50 centimeters per second H : 50 centimeters per second
0
0
or (9-2)
: 50 centimeters per second
1
H 1 H : 50 centimeters per second
It is important to remember that hypotheses are always statements about the population or
distribution under study, not statements about the sample. The value of the population param-
eter specified in the null hypothesis (50 centimeters per second in the above example) is usu-
ally determined in one of three ways. First, it may result from past experience or knowledge
of the process, or even from previous tests or experiments. The objective of hypothesis testing
then is usually to determine whether the parameter value has changed. Second, this value may
be determined from some theory or model regarding the process under study. Here the objec-
tive of hypothesis testing is to verify the theory or model. A third situation arises when the
value of the population parameter results from external considerations, such as design or en-
gineering specifications, or from contractual obligations. In this situation, the usual objective
of hypothesis testing is conformance testing.
A procedure leading to a decision about a particular hypothesis is called a test of a
hypothesis. Hypothesis-testing procedures rely on using the information in a random sample
from the population of interest. If this information is consistent with the hypothesis, we will con-
clude that the hypothesis is true; however, if this information is inconsistent with the hypothesis,
we will conclude that the hypothesis is false. We emphasize that the truth or falsity of a particu-
lar hypothesis can never be known with certainty, unless we can examine the entire population.
This is usually impossible in most practical situations. Therefore, a hypothesis-testing procedure
should be developed with the probability of reaching a wrong conclusion in mind.
The structure of hypothesis-testing problems is identical in all the applications that we
will consider. The null hypothesis is the hypothesis we wish to test. Rejection of the null
hypothesis always leads to accepting the alternative hypothesis. In our treatment of hypothe-
sis testing, the null hypothesis will always be stated so that it specifies an exact value of the
parameter (as in the statement H : 50 centimeters per second in Equation 9-1). The
0
alternate hypothesis will allow the parameter to take on several values (as in the statement
: 50 centimeters per second in Equation 9-1). Testing the hypothesis involves taking
H 1
a random sample, computing a test statistic from the sample data, and then using the test
statistic to make a decision about the null hypothesis.