There is a plus-four method for comparing difference in proportions. To get the plus four confidence interval for the difference p1p2, add four imaginary observations: one success and one failure in each of the two samples. Then use the large-sample confidence interval with the new sample sizes (actually sample sizes +2) and counts of successes (actual counts +1). This method can be used with any counts of successes and failures for each sample. Here is an example:The Abecedarian Project was a randomized controlled study to assess the effects of intensive early child-hood education on children who were at high risk based on several sociodemographic indicators. The project randomly assigned children to a treatment group, which was provided with early educational activities before kindergarten, and the remainder to a contol group. A recent follow-up study interviewed subjects at age 30and compared the college graduation rates. Here are the data for two groups :

Table 1

Population : 1

Population Description : treatment

Sample Size : n1=52

Number of Successes : 12

Sample Proportion : ^p1= 12/52 = 0.2308

Population : 2

Population Description : control

Sample Size : n2=49

Number of Successes : 3

Sample Proportion : ^p2= 3/49 = 0.0612

Question 1 : How much does early childhood education increase the proportion earning a four-year degree by age 30? Note that the there are only three successes in the control group so it is not ideal to use the traditional method. We can use the plus-four method here and the new data summary is :

Table 2

Population : 1

Population Description : treatment

Sample Size : n1+ 2 = 54

Number of Successes : 12+1 = 13

Sample Proportion : ~p1= 13/54 = 0.2407

Population : 2

Population Description : control

Sample Size : n2+ 2+ 51

Number of Successes : 3+1 = 4

Sample Proportion : ~p2= 4/51 = 0.0784

Question 2 : Compute the 90% confidence interval using Table 1.