Page 335 - Applied statistics and probability for engineers

Section 9-1/Hypothesis Testing 313

does not necessarily mean that there is a high probability that H 0 is true. It may simply mean
that more data are required to reach a strong conclusion. This can have important implications
for the formulation of hypotheses.
A useful analog exists between hypothesis testing and a jury trial. In a trial, the defendant
is assumed innocent (this is like assuming the null hypothesis to be true). If strong evidence is
found to the contrary, the defendant is declared to be guilty (we reject the null hypothesis). If evi-

dence is insuficient, the defendant is declared to be not guilty. This is not the same as proving
the defendant innocent and so, like failing to reject the null hypothesis, it is a weak conclusion.
An important concept that we will use is the power of a statistical test.
Power
The power of a statistical test is the probability of rejecting the null hypothesis H 0
when the alternative hypothesis is true.

The power is computed as 1−• , and power can be interpreted as the probability of correctly
rejecting a false null hypothesis. We often compare statistical tests by comparing their power
properties. For example, consider the propellant burning rate problem when we are testing
H 0 : = 50 centimeters per second against H 1 : ≠ 50 centimeters per second. Suppose that the
μ
μ
0
true value of the mean is μ = 52. When n = 10, we found that β = .2643, so the power of this
test is 1−• = − . 0 7357 when μ = 52.
1 0 2643= .
Power is a very descriptive and concise measure of the sensitivity of a statistical test when by
sensitivity we mean the ability of the test to detect differences. In this case, the sensitivity of the
test for detecting the difference between a mean burning rate of 50 centimeters per second and
52 centimeters per second is 0.7357. That is, if the true mean is really 52 centimeters per second,
μ
this test will correctly reject H 0 : = 50 and “detect” this difference 73.57% of the time. If this
value of power is judged to be too low, the analyst can increase either α or the sample size n.

9-1.3 One-Sided and Two-Sided Hypotheses
In constructing hypotheses, we will always state the null hypothesis as an equality so that the

probability of type I error α can be controlled at a speciic value. The alternative hypothesis might
be either one-sided or two-sided, depending on the conclusion to be drawn if H is rejected. If
0
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 alternative should be used.

Example 9-1 Propellant Burning Rate 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 conclu-
sion. The hypotheses should be stated as
H 0 : μ = 50 centimeters per second H 1 :μ , 50 centimeters per second
Here the critical region lies in the lower tail of the distribution of X. Because the rejection of H 0 is always a strong
conclusion, this statement of the hypotheses will produce the desired outcome if H 0 is rejected. Notice that, although

the null hypothesis is stated with an equals sign, it is understood to include any value of μ not speciied by the alter-
native hypothesis (that is, μ ≤ 50). Therefore, failing to reject H 0 does not mean that μ = 50 centimeters per second
exactly, but only that we do not have strong evidence in support of H 1 .

In some real-world problems in which one-sided test procedures are indicated, selecting
an appropriate formulation of the alternative hypothesis is occasionally difi cult. For example,
suppose that a soft-drink beverage bottler purchases 10-ounce bottles from a glass company.

The bottler wants to be sure that the bottles meet the speciication on mean internal pressure or