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. The hypotheses to be tested are H0: p = 0.65 versus Ha: p > 0.65. Clearly define the parameter p in the context of the problem.
b. Report the observed value of the test statistic for testing the hypotheses in part (a).
c. Compute the p-value for this test.
d. If the null hypothesis is true, what is the distribution of the test statistic (that is, which distibution should be used to find the p-value)?
- N(0,1) distribution
- N(0.65, 0.11) distribution
- Bin(n=20, p=0.65) distribution
- Bin(n-20, p=0.90) distribution
e. Our decision at the 5% level of significance is to reject the null hypothesis. Based on our decision, the appropriate conclusion in context is:
- There is not sufficient evidence to conclude that the sample proportion of cars with some damage exceeds the park's claim.
- There is not sufficient evidence to conclude that the population proportion of cars with some damage exceeds the park's claim.
- There is sufficient evidence to conclude that the sample proportion of cars with some damage exceeds the park's claim.
- There is sufficient evidence to conclude that the population proportion of cars with some damage exceeds the park's claim.
f. If the study will be repeated in the future with the same sample size selected from the sample populaton, but now a 10% significance level α will be used, then the probability of correctly rejecting the null hypothesis would:
- increase
- decrease stay the same