p-Values

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 μ₀ = 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 2P(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).