USING PROOF BY CONTRADICTION TO DRAW CONCLUSIONS 49 statistics related to the t- distribution, the z- distribution, the F- distribution, and the 2 -distribution. The p- value is particularly useful because it cuts across theoretical distributions. It is often referenced in publications fol- lowing a research conclusion, shown in parentheses to communicate the strength of the evidence supporting the research conclusion. In a one-tailed hypothesis test, the p- value is the amount of area beyond the test statistic into the tail of the rejection region. To find it on the z- distribution, we simply look the test statistic up on the Cumulative Stan- dard Normal Table. If the p- value is smaller than , then the test statistic itself falls beyond the critical bound of the rejection region. When that hap- pens, the null hypothesis is rejected. If the p- value is larger than , then the test statistic falls between the population parameter and the boundary of the rejection region. When that happens, the null hypothesis cannot be rejected. In a two-tailed hypothesis test, sample data are just as likely to be posi- tive as negative. So the amount of area beyond the test statistic into the tail of the distribution is only half of the p- value. To get the full p- value, the area beyond the test statistic has to be doubled. Comparisons of the p- value