AN ANALOGY: COIN FLIPS

Suppose someone comes up to you, hands you a coin, and says, "Tell me if this is a fair coin ” a coin for which heads and tails are equally likely." If you had nothing better to do, you would probably start flipping the coin and counting the number of times heads and tails occur. But you are no longer naive. You know that if you flip a fair coin 10 times, you will not often get exactly five heads and five tails. All sorts of outcomes are possible. There is even a reasonable chance that with a fair coin you will get eight tails and two heads or eight heads and two tails . However, as the coin becomes less and less unfair, it gets harder and harder for you to detect the difference. If the true probability of a head on the coin is .4999 instead of .5000, you would never figure that out unless you are willing to spend the rest of your life flipping coins . Any combination of flips that you come up with will appear perfectly reasonable if the coin is fair or if it is minutely biased . Although you can disprove with a certain degree of confidence that a coin is fair, it is impossible to prove that it is exactly fair. That is why we cannot say we proved that the average of any variable or group is the same as the average of another variable or group . All we can say is that the evidence did not disprove it.