Download Introduction to Chance Models (Section 1.1) Introduction A key step

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The sample size in this example is 16. Note that the word “sample” is used as both a noun (the
set of observational units being studied) and as an adjective, for example to mean “computed
from the observed data,” as, for example, “sample statistic.”
Step 4: Draw inferences beyond the data. These 16 observations are a mere snapshot of
Buzz’s overall selection process. We will consider this a random process. We are interested in
Buzz’s actual long-run probability of pushing the correct button based on Doris’ 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 can’t 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:


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 can’t collect more data,
we have to base our conclusions only on the data we have. It’s 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 don’t 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 “Buzz was simply guessing” 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 Buzz’s choice assuming he is just guessing which button to push. To
June 27, 2014
MAA PREP workshop
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