Comparing the critical value and p-value methods
Thus far we have considered two methods for testing H0: μ = 50 versus Ha: μ > 50 at the .05 and .01 values of α. Using the first method, we determine if the test statistic value z = 2.20 is greater than the critical values z.05 = 1.645 and z.01 = 2.33. Using the second method, we determine if the p-value of .0139 is less than .05 and .01. Whereas the critical value method requires that we look up a different critical value for each different α value, the p-value method requires only that we calculate a single p-value and compare it directly with the different α values. It follows that the p-value method is the most efficient way to test a hypothesis at different α values. This can be useful when there are different decision makers who might use different α values. For example, television networks do not always evaluate advertising claims by setting α equal to .05. The reason is that the consequences of a Type I error (advertising a false claim) are more serious for some claims than for others. For example, the consequences of a Type I error would be fairly serious for a claim about the effectiveness of a drug or for the superiority of one product over another. However, these consequences might not be as serious for a noncomparative claim about an inexpensive and safe product, such as a cosmetic. Networks sometimes use α values between .01 and .04 for claims having more serious Type I error consequences, and they sometimes use α values between .06 and .10 for claims having less serious Type I error consequences. Furthermore, one network’s policies for setting α can differ somewhat from those of another. As a result, reporting an advertising claim’s p-value to each network is the most efficient way to tell the network whether to allow the claim to be advertised. For example, most networks would evaluate the trash bag claim by choosing an α value between .025 and .10. Since the p-value of .0139 is less than all these α values, most networks would allow the trash bag claim to be advertised.