At one point during the 2015 NFL season, head coach Bill Belichick and the New England Patriots had won 19 of their past 25 called coin flips at the beginning of NFL games (?For Bill Belichick, Patriots? Strategy Is No Flip of theCoin,? bostonglobe.com/sports/2015/11/04? /pnotes/vFNt235bsK8x3JLZ6FJdtK/story.html, November 4, 2015, retrieved May 6, 2017). Suppose that these 25 coin toss calls can be considered as representative of all coin toss calls made by this team.

a. Perform an exact binomial test to determine if there is convincing evidence that the proportion of all coin flip calls that the Patriots win is greater than 0.5.

b. Discuss the conditions required for the exact binomial version of the hypothesis test. Write a brief explanation of why the results of the test performed in Part (a) do not necessarily mean that Coach Belichick is able to predict the results of coin flips better than other coaches.

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Randomization Distribution 300 200 100 0.2 0.3 Sample Proportion 0.1 0.4 0.5 0.6 Observed Sample Proportion Null Hypothesis Standard Error P-Value # of Simulated Samples # of Values in Tail 1000 0.680 0.100 0.059 0.000 Frequency Binomial Distribution 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 50 100 150 Successes 200 250 300 Observed Sample Proportion Null Hypothesis P-Value 0.661 0.500 0.000 Probability