What is the fraction of sample means expected to fall within the 95% confidence intervals? This is an incorrect (and not useful) interpretation of the 95% CI, but out of curiosity, the answer is less than 95%.

CI_confusion.R

fracwithin <- function(n){
  sam1 <- rnorm(n)
  ci <- t.test(sam1)$conf.int
  means <- replicate(100, mean(rnorm(n)))
  mean(means > ci[1] & means < ci[2])
}

set.seed(2)
n <- seq(5, 10000, by=2)
fraction <- sapply(n, function(i) fracwithin(i))

plot(n, fraction)
abline(h=0.95, lw=4)
lines(smooth.spline(n, fraction), col="red", lw=5, lty=3)
legend("bottomright", col = c("black", "red"), lty=c(1, 3),
       legend=c("0.95", "frac sample means within 95% CI"))