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FAQ 1.1: What is a random process? Q: So a parameter is a numerical property of a processâ¦ but what is the process? All I have is Buzzâs 16 attempts. A: We can think of Buzzâs 16 attempts as a sample from his long-run selection process. He has some underlying probability of pushing the correct button, but heâs unlikely to be correct every time, even if the dolphins are communicating. There is randomness in his choices. His choices might be affected by Dorisâ squeaks, but also by how hungry he is, how tired he is, how warm the water is. We are trying to see how much Dorisâ communication is influencing him. Everything else gets lumped together into ârandom chance.â Any one outcome from a random process is unknown to us in advance (like tossing a coin), but if we observe outcomes from the process for long enough, we should start to see some patterns amidst the randomness. So you can think of a random process as an unlimited source of potential observations for your sample. Q: Can I use any old set of observations from a process as a sample? A: Some samples are good and others are terrible. Q: How do I tell the good apples from the rotten ones? A: For a Yes/No process like Buzzâs selections, we need to be willing to make certain assumptions along the way. For example, we assumed Buzzâs probability of a correct guess was the same every time. In fact, under the null hypothesis, we assumed it was 50/50 every time. But even if the probability is not 0.5, we are assuming it is not changing. In other words, we assume that Buzz doesnât get better through learning and that he doesnât get tired or bored or quit trying. We are also assuming that his previous trials donât influence his future guesses â like a coin, he would have no memory of his past guesses. Q: So how do I know if I have a good sample? A: You need to believe that nothing about the process is changing and that each outcome does not depend on past outcomes. If youâre a coin, your chance of heads doesnât change, and you have no memory of how youâve landed in the past. If this is true about the process, then you will have a good sample! On the other hand, if Buzzâs probability of success is different in the morning than in the afternoon, then only observing him in the morning will not give you a good representation of his overall probability of success. Q: So then how does the parameter come into play? A: Parameter is a hard concept, precisely because we never see it! For the dolphin example, we want to know whether Buzz is just guessing. We think of his guesses as a potentially never-ending process (like coin tossing), and the parameter is the probability that Buzz will be correct in his next attempt. As you saw in the Preliminaries, this means if we were to observe the random process forever, this probability is the long-run proportion of times that Buzz pushes the correct button. Q: So I see why the parameter isnât just 15/16, that is his sample proportion and we may not have hit that long-run probability yet. So the parameter is just 0.5? A: Not necessarily. The parameter equals 0.5 if he is just guessing. Thatâs the chance model we simulated. But his probability may actually be different from 0.5. Q: So the parameter is his actual probability of pushing the correct button. Itâs a number, but we donât know its value. A: Correct! But keep in mind that we will define lots of different kinds of parameters in future chapters. The âprobability of successâ parameter only applies in this type of Yes/No random process. June 27, 2014 MAA PREP workshop 18