Final answer:
Option 2, giving participants caffeine-free coffee, is the best control for testing the impact of caffeine on math test performance, as it accounts for psychological and physical variables.
Step-by-step explanation:
Artin's best control group to test the theory that drinking caffeine before a math test improves performance would be option 2: Participants picked from the same population are given 500 mL of caffeine-free coffee 30 minutes before the test. This option provides the most controlled environment by accounting for the possible placebo effect and other variables like taste and volume of liquid consumed. It is crucial to control for these factors, as results from other studies indicate that believing one has taken caffeine can affect performance, and consuming caffeine without knowledge doesn't necessarily lead to significant performance increments.