CI "precision" reversal

precision.R

desired_precision = .6

N1 = 5
N2 = 6

# Function to compute CDFs of "precisions" (CI widths)
SS_func = function(W, n, alpha = .05){
  n * (n - 1) * W^2 / (4 * qt(1-alpha/2, n - 1)^2)
}

# Show theoretical CDFs
curve( pchisq(SS_func(x, N1), N1 - 1), 0, 6, 256, col = "blue")
curve( pchisq(SS_func(x, N2), N2 - 1), 0, 6, 256, col = "red", add = TRUE)

# Confirm through simulation
widths_n1 = replicate(100000, {
  diff(t.test(rnorm(N1))$conf.int)
})

widths_n2 = replicate(100000, {
  diff(t.test(rnorm(N2))$conf.int)
})

## Add simulations to plots
## Should exactly overlay theoretical curves
plot(ecdf(widths_n1), add = TRUE, col = "blue", lwd = 2)
plot(ecdf(widths_n2), add = TRUE, col = "red", lwd = 2)


# Zoom in
curve( pchisq(SS_func(x, N1), N1 - 1), 0, 2*desired_precision, 256, col = "blue")
curve( pchisq(SS_func(x, N2), N2 - 1), 0, 2*desired_precision, 256, col = "red", add = TRUE)
abline(v = desired_precision)

## Notice that the blue line (smaller N) is *above* the red line (larger N) at 
## the desired precision, indicating that there is a greater probability that 
## the "precision" (CI width) is smaller than the desired precision is 
## *larger* for for the smaller N.