Two Groups, A & B, undergo same intervention. Group A (n=20). Group B (n=10). The only outcomes are symptomatic or asymptomatic. After the intervention, 4/20 people in group A are asymptomatic. In Group B 6/10 are asymptomatic. Is this statistically significant (i.e. p <0.05).
Rate A is
4/20 = 0.2. Rate B is
6/10 = 0.6. By "Is this statistically significant" I presume you mean, "is the difference between rate A and rate B statistically significant?" (And not "is the treatment effective?")
The implied null hypothesis is that the two groups have the same underlying rate, and that the observed difference was simply due to chance. You can test the equality of two rates in R as follows:
AB=matrix(c(4,16,6,4),ncol=2,byrow=T) #4+16=20 and 6+4=10 rownames(AB)=c("A","B") colnames(AB)=c("asym","sym") # Take a look at the table: AB # We can use a chi-squared test, first using Yates' continuity correction, prop.test(AB) #p = 0.075 # and also try it without Yates' continuity correction: prop.test(AB, correct = FALSE) #p = 0.028 # If your data has fixed marginals (i.e., the category counts 10 and 20 # were not random elements in the experimental design), then the # Fisher exact test may be the best way to do this: fisher.test(AB) #p = 0.04486
In summary, I've confirmed your result.
Edit: This answer has been rewritten after some horrendous mistakes pointed out by @Glen_b.