The p-value is the probability of the occurrence of a value for the test statistic as extreme as or more extreme than the actual observed value, under the assumption that the null hypothesis is true.

*p***-VALUES**

The *p*-value
is the probability of the occurrence of a value for the test statistic as
extreme as or more extreme than the actual observed value, under the
assumption that the null hypothesis is true. By more extreme we mean a value in
a direction farther from the center of the sampling distribution (under the
null hypothesis) than what was observed.

For a one-tailed (right-tailed) *t* test, this statement means the
probability that a statistic *T* with a
Student’s *t* distribution satisfies *T* > |*t*|, where *t* is the
observed val-ue of the test statistic. For a one-tailed (left-hand tail) *t* test, this statement means the
probability that a statistic *T* with a
Student’s *t* distribution satisfies *T* < –|*t*|, where *t* is the
observed value of the test statistic. For a two-tailed *t* test, it means the probability that a statistic *T* with a Student’s *t* distribution satisfies |*T*|
> |*t*| (i.e., *T* > |*t*| or *T* < –|*t*|) where *t* is the
observed value of the test statistic.

Now let us now compute the two-sided *p*-value for the test statistic in the
pig blood loss example from Section 9.4. Recall that the standard deviation s =
717.12, the sample mean =
1085.9, the hypothesized value μ_{0} = 2200, and the sample size *n *=
10. From this information, we see that the*
t *statistic is* t *= (1085.9 –* *2200)/(717.12/√10) = –1114.1/226.773 = –4.913.

To find the two-sided *p*-value we must compute the probability that *T* > 4.913 and add the probability that *T* < –4.913. This combination is equal to 2*P*(*T* > 4.913). The
probability *P*(*T* > 4.913) is the one-sided right-tail *p*-value; it is also equal to the one-sided left-tail *p*-value, *P*(*T* < –4.913). The
table of Student’s *t* distribution
shows us that with 9 degrees of freedom, *P*(*T* < 4.781) = 0.9995. So *P*(*T*
> 4.781) = 0.0005.

Since *P*(*T* > 4.913) < *P*(*T* > 4.781), we see
that the one-sided *p*-value *P*(*T*
> 4.913) < 0.0005; hence, the two-sided *p*-value is less than 0.001. This observation is more informative
than just saying that the test is significant at the 5% level. The re-sult is
so significant that even for a two-sided test, we would reject the null
hypoth-esis at the 0.1% level.

Most standard statistical packages (e.g., SAS)
present *p*-values when providing
information on hypothesis test results, and major journal articles usually report
*p*-values for their statistical tests.
SAS provides *p*-values as small as
0.0001, and any-thing smaller is reported simply as 0.0001. So when you see a *p*-value of 0.0001 in SAS output, you
should interpret it to mean that the *p*-value
for the test is actually less than or equal to 0.0001 (sometimes it can be
considerably smaller).

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