Incentives for Quitting Smoking: Do They Work? Exercise 5.31 describes a study examining incentives to quit smoking. With no incentives, 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.

(a) State the null and alternative hypotheses, and give the notation and value of the sample statistic.

(b) Use a randomization distribution and the observed sample statistic to find the \(p\)-value.

(c) 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.

(d) Use the standard error found from the randomization distribution in part (b) to find the standardized test statistic, and then use that test statistic to find the p-value using a standard normal distribution.

(e) Compare the p-values from parts (b), (c), and (d). Use any of these p-values to give the conclusion of the test.

Data From Exercise 5.31:
Incentives for Quitting Smoking: Group or Individual? 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. After six months in the program, 148 of the 1080 in the group program were successfully abstaining from smoking, while 120 of the 990 in the individual program were successful. We wish to test to see if this data provide evidence of a difference in the proportion able to quit smoking between smokers in a group program and smokers in an individual program.