Determining regions of rejection The diligent researcher considers all 6-tuples of success and failure as the sample space, and takes their numbers as sufficient statistic. The event of six successes, or six correct guesses, has a probability of \(1 / 2^{6} = 1/64\) under the null hypothesis that the lady is merely guessing, against a probability of \(3^{6} / 4^{6}\) under the alternative hypothesis. If we set \(r &lt; 3^{6} / 2^{6}\), then this sample is included in the region of rejection of the null hypothesis. Samples with five successes have a probability of \(1/64\) under the null hypothesis too, against a probability of \(3^5 / 4^{6}\) under the alternative. By lowering the likelihood ratio by a factor 3, we include all these samples in the region of rejection. But this will lead to a total probability of false rejection of \(7/64\), which is larger than 5%. So these samples cannot be included in the region of rejection, and hence the diligent researcher does not reject the null hypothesis upon finding five successes and one failure.