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continue our repetitions of 16 tosses, we can start to see how the outcomes for the number of heads are distributed. Does the distribution of the number of heads that result in 16 flips have a predictable long-run pattern? In particular, how much variability is there in our simulated statistics between repetitions (sets of 16 flips) just by random chance? In order to investigate these questions, we need to continue to flip our coin to get many, many sets of 16 flips (or many repetitions of the 16 choices where we are modeling Buzz simply guessing each time). We did this, and Figure 1.4 shows what we found when we graphed the number of heads from each set of 16 coin flips. Here, the process of flipping a coin 16 times was repeated 1,000 timesâthis number was chosen for convenience, but also appears to be large enough to give us a fairly accurate sense of the long-run behavior for the number of heads in 16 tosses. Figure 1.4: A dotplot showing 1000 repetitions of flipping a coin 16 times and counting the number of heads Each dot represents one set of 16 attempts by Buzz Letâs think carefully about what the graph in Figure 1.4 shows. For this graph, each dot represents the number of heads in one set of 16 coin tosses. We see that the resulting number of heads follows a clear pattern: 7, 8, and 9 heads happened quite a lot, 10 was pretty common also (though less so than 8), 6 happened some of the time, 1 happened once. But we never got 15 heads in any set of 16 tosses! We might consider any outcome between about 5 and 11 heads to be typical, but getting fewer than 5 heads or more than 11 heads happened rarely enough we can consider it a bit unusual. We refer to these unusual results as being out in the âtailsâ of the distribution. Think about it: How does the analysis above help us address the strength of evidence for our research conjecture that Buzz was doing something other than just guessing? What does this have to do with the dolphin communication study? We said that we would flip a coin to simulate what could happen if Buzz was really just guessing each time he pushed the button in 16 attempts. We saw that getting results like 15 heads out of 16 never happened in our 1000 repetitions. This shows us that 15 is a very unusual outcomeâ far out in the tail of the distribution of the simulated statisticsâif Buzz is guessing. In short, even though we expect some variability in the results for different sets of 16 tosses, the pattern shown in this distribution indicates that an outcome of 15 heads is outside the typical chance variability we would expect to see when Buzz is simply guessing. June 27, 2014 MAA PREP workshop 5