If the number of success or the number of failures is less than 10, then the confidence interval has a far less success rate in capturing the population parameter. Fortunately, there is a simple alternative procedure, know as "plus-four", that works remarkably well. The idea of the plus-four interval is to pretend that the sample contained two more "successes" and two more"failures" than it actually did, and then carry on like always. If the sample proportion is p=x/n where x represents the number of success and n represent the sample size. Then the modified sample proportion after including the fictional successes and failures is denoted by

p= (x+ 2)/(n+ 4)

The plus-four 95% confidence interval is therefore:

p 1.96 . p(1p) / n+ 4 (SQUARE ROOT)

Question (1) Let p= 3/10. Compute p.

Question (2) Let p= 8/13. Compute p.

Question (3) In general, how does p compare to p?

Question (4) Does this fairly simple plus-four adjustment really fix the problem? Let us find out by simulating again. Go to the following applethttp://www.rossmanchance.com/applets/ConfSim.html, under method, choose proportions, binomial and Wald. Let us choose the sample size n= 15, and a population proportion p= 0.1 (in the applet you will see instead of p so choose = 0.1). Choose intervals to be 500 and the confidence interval to be 95%. Hit "Sample". What is the percentage of the intervals containing 0.1? (Include a screenshot) and now choose (Plus Four 95% instead of Wald), keep the rest the same. What is the new percentage of the intervals containing 0.1? (Include a screenshot)