In a smoking cessation program, over 2000 smokers who were trying to quit were randomly assigned to either a group program or an individual program. With no incentive, the proportion of smokers trying to quit who are still abstaining six months later is about 0.06. Participants in the study were randomly assigned to one of four different incentives, and the proportion successful was measured six months later. Of the 498 participants in the group with the least success, 47 were still abstaining from smoking six months later. We wish to test to see if this provides evidence that even the smallest incentive works better than the proportion of 0.06 with no incentive at all.

Required:
a. State the null and alternative hypotheses.
b. Give the notation for the sample statistic.
c. Find the value of the sample statistic.
d. Use a randomization distribution and the observed sample statistic to find the p-value.
e. Give the mean and standard error of the normal distribution that most closely matches the randomization distribution, and then use this normal distribution with the observed sample statistic to find the p-value.
f. Use the standard error found from the randomization distribution in part (d) to find the standardized test statistic, and then use that test statistic to find the p-value using a standard normal distribution.
g. Compare the p-values from the previous parts of the question What is the conclusion?