I'm using R to calculate the two-sample test for equality of proportions, where the two proportions are 350/400 and 25/25. So:
> prop.test(c(350,25),c(400,25)) 2-sample test for equality of proportions with continuity correction data: c(350, 25) out of c(400, 25) X-squared = 2.4399, df = 1, p-value = 0.1183 alternative hypothesis: two.sided 95 percent confidence interval: -0.17865986 -0.07134014 sample estimates: prop 1 prop 2 0.875 1.000 Warning message: In prop.test(c(350, 25), c(400, 25), correct = FALSE) : Chi-squared approximation may be incorrect
What I can't reconcile on my own is that the p-value is greater than 0.05, and yet the 95% confidence interval for the difference does not include 0. I thought there was an 'if and only if' relationship between the two (The p-value < alpha iff the (1-alpha) confidence interval of the difference does not include 0).
What am I not seeing? My only guess is there's something fundamental that I'm misunderstanding, or that it has something to do with that warning message about chi-squared approximation.