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The sample size in this example is 16. Note that the word âsampleâ is used as both a noun (the set of observational units being studied) and as an adjective, for example to mean âcomputed from the observed data,â as, for example, âsample statistic.â Step 4: Draw inferences beyond the data. These 16 observations are a mere snapshot of Buzzâs overall selection process. We will consider this a random process. We are interested in Buzzâs actual long-run probability of pushing the correct button based on Dorisâ whistles. This unknown long-run probability is called a parameter. Definition: For a random process, a parameter is a long-run numerical property of the process. Note that we are assuming this parameter is not changing over time, at least for the process used by Buzz in this phase of the study. Because we canât observe Buzz pushing the button forever, we need to draw conclusions (possibly incorrect, but hopefully not) about the value of the parameter based only on these 16 attempts. Buzz certainly pushed the correct button most of the time, so we might consider either: ï· ï· Buzz is doing something other than just guessing (his probability of a correct button push is larger than 0.50). Buzz is just guessing (his probability of a correct button push is 0.50) and he got lucky in these 16 attempts. These are the two possible explanations to be evaluated. Because we canât collect more data, we have to base our conclusions only on the data we have. Itâs certainly possible that Buzz was just guessing and got lucky! But does this seem like a reasonable explanation to you? How would you argue against someone who thought this was the case? Think about it: Based on these data, do you think Buzz somehow knew which button to push? Is 15 out of 16 correct pushes convincing to you? Or do you think that Buzz could have just been guessing? How might you justify your answer? So how are we going to decide between these two possible explanations? One approach is to choose a model for the random process (repeated attempts to push the correct button) and then see whether our model is consistent with the observed data. If it is, then we will conclude that we have a reasonable model and we will use that model to answer our questions. The Chance Model Scientists use models to help understand complicated real world phenomena. Statisticians often employ chance models to generate data from random processes to help them investigate such processes. You did this with the Monty Hall Exploration (P.3) to investigate properties of the two strategies, switching and staying with your original choice of door. In that exploration it was clear how the underlying chance process worked, even though the probabilities themselves were not obvious. But here we donât know for sure what the underlying real world process is. We are trying to decide whether the process could be Buzz simply guessing or whether the process is something else, such as Buzz and Doris being able to communicate. Let us first investigate the âBuzz was simply guessingâ process. Because Buzz is choosing between two options, the simplest chance model to consider is a coin flip. We can flip a coin to represent or simulate Buzzâs choice assuming he is just guessing which button to push. To June 27, 2014 MAA PREP workshop 3