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292 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

9-2.2 P-Values in Hypothesis Tests

One way to report the results of a hypothesis test is to state that the null hypothesis was or was
not rejected at a specified -value or level of significance. For example, in the propellant
problem above, we can say that H : 50 was rejected at the 0.05 level of significance. This
0
statement of conclusions is often inadequate because it gives the decision maker no idea about
whether the computed value of the test statistic was just barely in the rejection region or
whether it was very far into this region. Furthermore, stating the results this way imposes the
predefined level of significance on other users of the information. This approach may be un-
satisfactory because some decision makers might be uncomfortable with the risks implied by
0.05.
To avoid these difficulties the P-value approach has been adopted widely in practice.
The P-value is the probability that the test statistic will take on a value that is at least as
is true. Thus, a
extreme as the observed value of the statistic when the null hypothesis H 0
P-value conveys much information about the weight of evidence against H , and so a deci-
0
sion maker can draw a conclusion at any specified level of significance. We now give a
formal definition of a P-value.

Definition
The P-value is the smallest level of significance that would lead to rejection of the
null hypothesis H 0 with the given data.

It is customary to call the test statistic (and the data) significant when the null hypoth-
is rejected; therefore, we may think of the P-value as the smallest level at which
esis H 0
the data are significant. Once the P-value is known, the decision maker can determine how
significant the data are without the data analyst formally imposing a preselected level of
significance.
For the foregoing normal distribution tests it is relatively easy to compute the P-value. If
z is the computed value of the test statistic, the P-value is
0

0
0
1
231
1|z 0 |24 for a two-tailed test: H : H : 0
P •1
1z 2 for a upper-tailed test: H : H : 0 (9-15)
1
0
0
0
1z 2 for a lower-tailed test: H : H : 0
0
0
1
0
Here, 1z2 is the standard normal cumulative distribution function defined in Chapter 4.
Recall that 1z2 P1Z z2 , where Z is N(0, 1). To illustrate this, consider the propellant
problem in Example 9-2. The computed value of the test statistic is z 3.25 and since the
0
alternative hypothesis is two-tailed, the P-value is
P-value 231
13.2524 0.0012
Thus, H : 50 would be rejected at any level of significance P-value 0.0012. For
0
example, H would be rejected if 0.01 , but it would not be rejected if 0.001 .
0
It is not always easy to compute the exact P-value for a test. However, most modern
computer programs for statistical analysis report P-values, and they can be obtained on some
hand-held calculators. We will also show how to approximate the P-value. Finally, if the