This chapter continues our development of inference to a new situation, one in which the objective is to choose between two alternatives. One alternative is thought to be most likely to apply and is labeled the “null hypothesis,” and the other is labeled the “alternative hypothesis.” We ask questions such as, “Does process A take longer than B?” “Is the probability of contracting polio more likely using live antibodies or dead ones?” “Do daddy longlegs learn from their mistakes in losing their legs?” or “Are sunspots a source of global warming?” We restrict our attention to just two alternatives. This inference procedure is called “hypothesis testing.” We demonstrate how one can devise a strategy to use observed data to choose between the two alternatives in an “optimal manner.” In setting up the hypothesis test, we note that there are two states of the world, we have only two choices to make, and that in any situation we can make one of two errors. Either we incorrectly reject the null, or we incorrectly reject the alternative hypothesis. To choose an optimal test procedure, we need to assess the cost, or importance to us, of each of these errors. In the process, we clarify the importance and limitations of hypothesis testing. We also introduce the idea of a P value as a measure of the likelihood of the observed value of the statistic under the “null,” or the presumed, hypothesis. In this way, we can transfer information expeditiously to others.