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286 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

9-1.3 One-Sided and Two-Sided Hypotheses

A test of any hypothesis such as

H 0 : 0
H : 0
1
is called a two-sided test, because it is important to detect differences from the hypothesized value
of the mean 0 that lie on either side of 0 . In such a test, the critical region is split into two parts,
with (usually) equal probability placed in each tail of the distribution of the test statistic.
Many hypothesis-testing problems naturally involve a one-sided alternative hypothesis,
such as

H : 0 H : 0
0
0
H : 0 or H : 0
1
1
, the critical region should lie in the upper tail
If the alternative hypothesis is H : 0
1
of the distribution of the test statistic, whereas if the alternative hypothesis is H : ,
1
0
the critical region should lie in the lower tail of the distribution. Consequently, these tests
are sometimes called one-tailed tests. The location of the critical region for one-sided tests
is usually easy to determine. Simply visualize the behavior of the test statistic if the null
hypothesis is true and place the critical region in the appropriate end or tail of the distri-
bution. Generally, the inequality in the alternative hypothesis “points” in the direction of
the critical region.
In constructing hypotheses, we will always state the null hypothesis as an equality so that
the probability of type I error can be controlled at a specific value. The alternative hypoth-
esis might be either one-sided or two-sided, depending on the conclusion to be drawn if H is
0
rejected. If the objective is to make a claim involving statements such as greater than, less
than, superior to, exceeds, at least, and so forth, a one-sided alternative is appropriate. If no
direction is implied by the claim, or if the claim not equal to is to be made, a two-sided alter-
native should be used.

EXAMPLE 9-1 Consider the propellant burning rate problem. Suppose that if the burning rate is less than
50 centimeters per second, we wish to show this with a strong conclusion. The hypotheses
should be stated as

: 50 centimeters per second
H 0
H 1 : 50 centimeters per second

Here the critical region lies in the lower tail of the distribution of . Since the rejection of H 0
X
is always a strong conclusion, this statement of the hypotheses will produce the desired out-
come if H is rejected. Notice that, although the null hypothesis is stated with an equal sign, it
0

is understood to include any value of not specified by the alternative hypothesis. Therefore,
failing to reject H does not mean that 50 centimeters per second exactly, but only that we
0
do not have strong evidence in support of H 1 .
In some real-world problems where one-sided test procedures are indicated, it is
occasionally difficult to choose an appropriate formulation of the alternative hypothesis. For
example, suppose that a soft-drink beverage bottler purchases 10-ounce bottles from a glass