Suppose your company had designed and created a memorial coin for the next Super Bowl. The "master" coin would be used for the ceremonial coin-flip to begin the game, and fans could order a replica of that coin for themselves. It's very important that the master coin be fair, that is, that the probability of heads of the master coin be 0.50.  The NFL requires a formal report as to the scientific assessment of that probability.

Suppose it was decided that once the master coin was created, a study would be conducted where the coin would be tossed on artificial turf 500 times and the number of heads would be recorded. The study plan is to test H0: p = 0.50  versus  Ha: p ≠ 0.50 at a 5% significance level.

Of these 500 tosses, 261 resulted in "heads" for a sample proportion of 0.522.  

a.  Assuming one can consider these tosses as random and independent, check the remaining condition that is needed in order to conduct a large sample z-test of the hypotheses.  

b. Compute the observed z test statistic value.  

c. Find the corresponding p-value for this test.  Include a complete sketch or graph of the p-value as part of your work, shading the p-value area and labeling the distribution and x-axis.

d. At a 10% significance level, the decision is to fail to reject the null hypothesis.  Which of the following is the appropriate conclusion? 

• There is insufficient evidence to show the coin is biased, so the master coin could be used in the ceremonial coin-flip.
• There is sufficient evidence to show the coin is biased, that is, the master coin should not be used in the ceremonial coin-flip.