Suppose that I have two approaches to a particular problem. Approach A is observed to succeed 685 times out of 1347 attempts. Approach B is observed to succeed 2100 times out of 3748 attempts. I want to see if Approach B is preferable to Approach A.
In R I run:
prop.test(c(2100,685), c(3748,1347), alternative="greater", correct=FALSE)
and I get:
data: c(2100, 685) out of c(3748, 1347) X-squared = 10.7124, df = 1, p-value = 0.0005321 alternative hypothesis: greater 95 percent confidence interval: 0.02568765 1.00000000 sample estimates: prop 1 prop 2 0.5602988 0.5085375
I have the following questions.
Let $p_A$ (resp. $p_B$) be the proportion of successes of Approach A (resp. B). Implicitly, my null hypothesis is that $p_A=p_B$, right? And then, since the $p$-value is less than the significance level, I reject the null hypothesis. To me this seems that I am led to conclude that $p_A \neq p_B$. Of course, it seems likely by just looking at the data that $p_B>p_A$.
Is the syntax
alternative="greater"included to rule out $p_A>p_B$?
Similarly, does the line about the confidence interval mean that the "true" value of $p_B-p_A$ lies in the interval $(.02568765,1)$?